LIBRARY OF CONGRESS. 



QA303 

Chap. Copyright No.. 



Sheli 

UNITED STATES OF AMERICA. 



CALCULUS 



WITH APPLICATIONS 



AN INTRODUCTION TO THE MATHEMATICAL 
TREATMENT OF SCIENCE 



BT 

ELLEN HAYES 

PROFESSOR OF APPLIED MATHEMATICS IN WELLESLEY COLLEGE 



Boston 

ALLYN AND BACON 

1900 



__ 70387 

i-itor^i y of Oonor«9« 

'\*u Cohu Hectiveo 
NOV 3 1900 

Copyright entry 

SECOND COPY. 

Delivered to 

ORDfcS DIVISION, 

NOV 20 1900 



COPYRIGHT, 19 00, 
BY ELLEN HAYES. 









Ncrfoooti -Iftes 

J. S. Cushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 



PREFACE. 

This little book lias been written for two classes of 
persons : those who wish, for purposes of culture, to know, 
in as simple and direct a way as possible, what the calculus 
is and what it is for; and students primarily engaged in 
work in chemistry, astronomy, economics, etc., who have 
not time or inclination to take long courses in mathematics, 
yet who would like " to know how to use a tool as fine as 
the calculus. " 

The 'pure' mathematician will note the omission of 

various subjects that are important from his point of view; 

but for him there are admirable and lengthy treatises on 

pure calculus. Also the student whose experience has 

led him to conceive of mathematical study as the doing 

of interminable lists of exercises, will be surprised and, 

possibly, disax->pointed. This book is a reading lesson in 

applied mathematics. Fancy exercises have been avoided. 

The examples are, for the most part, real problems from 

mechanics and astronomy. This plan has been pursued in 

the conviction that such problems are just as good as 

make-believe ones for purposes of discipline, and a good 

deal better for purposes of knowledge. The time-honored 

method of presenting calculus is much as if travelers 

should be stopped and made to pound stone on the high- 

iii 



IV PREFACE. 

way, so that they never get anywhere or even know what 
the road is for. The following pages are a protest against 
the conventional method ; for I am wholly in sympathy 
with a remark made by Professor Lester F. Ward, in his 
Outlines of Sociology : "There is no more vicions educa- 
tional practice, and scarcely any more common one, than 
that of keeping the student in the dark as to the end and 
purpose of his work. It breeds indifference, discourage- 
ment, and despair." 

A chapter on analytic geometry has been introduced, in 
the hope that teachers will try the plan of presenting the 
elements of the calculus and of analytic geometry together. 
There is no good reason either for keeping them distinct 
or for presenting analytic geometry first. 

To three works I have to express my deep obligation. 
The spirit manifest in them has been my chief encourage- 
ment in preparing this book. I refer to GreenhilPs Differ- 
ential and Integral Calculus, Perry's Calculus for Engineers, 
and Nernst and Schonnies' Einfulirung in die mathematische 
Behandlung der Naturwissenschaften. 

We have in these works, let us hope, an indication of 
the role which the calculus is to play in schemes for liberal 
and scientific education in the not far distant future. 



ELLEN HAYES. 



Wellesley College, 
September, 1900. 



CONTENTS. 



CHAPTER I. 
Differentiation and Integration. 

ARTICLES PAGE 

1- 6. Introduction 1 

7-13. Differentiation of algebraic functions ... 8 

1-4-16. Integration 12 

17. Exercises 14 

18-19. Implicit functions. Exercises 16 

20-22. Differentiation of trigonometric functions ... 17 
23-24. Differentiation of exponential and logarithmic func- 
tions 21 

25-27. Second derivatives. Partial and total differentials . 24 

28. Taylor's theorem 26 

29. Maclaurin's theorem ....... 28 

30-31. Binomial theorem. Converging series ... 29 

32-33. Indeterminate forms 31 

34-35. Exercises. Logarithms 34 

CHAPTER II. 
The Graph. 



36-37. Cartesian system of coordinates . 

dy 
38-40. Geometric meaning of ~ Exercises 

ax 

41-43. Maxima and minima. Exercises 

44. Examples in maxima and minima 

45-46. Polar coordinates .... 

v 



41 

44 

48 
50 
56 



VI CONTENTS. 

CHAPTER III. 
Applications. 

ARTICLES PAGE 

47-54. Velocity and acceleration 58 

55-59. Simple harmonic motion 64 

60-63. Falling bodies 67 

64-65. Rectilinear motion ....... 74 

66-69. Parabolic motion 77 

70. Motion in a vertical curve 81 

71-72. Simple pendulum 82 

73-74. Areas. Examples 85 

75-77. Mean values 88 

78-79. Work 91 

80. Lengths of curves 92 

81. Volumes and surfaces of revolution ... 94 

82. Double and triple integrals 96 

83. Perfect differential 97 

84-85. Moment of inertia. Examples .... 99 

86-93. Kepler's laws 102 

CHAPTER IV. 

Analytic Geometry. 

94-99. The equation / (x, y) = Ill 

114 
117 
117 
118 
118 
119 
119 
121 



100-103. Change of axes 

104. Condition of parallelism, and of perpendicularity 

105. Straight line in terms of slope and intercept . 

106. Straight line in terms of two points 

106. Straight line in terms of one point and slope . 

107. Distance between two points .... 

108. Distance from point to line .... 
109-116. The ellipse 



CONTENTS. vil 

ARTICLES PAGE 

117-125. The hyperbola 127 

126-131. The parabola 134 

132-133. Tangent and normal to a curve .... 137 

134. Path of middle point of ellipse-chord . . . 140 

135. Determination of center and axes of ellipse . . 142 

136. Tangent in terms of slope and intercept . . . 143 

137. Exercises 145 

138-139. Space coordinates .147 

140-141. Distance between two points in space . . . 148 

142. Equation to a plane surface 148 

CHAPTER Y. 
Formulas. 

Fundamental integrals . 151 

Other integrals . . . . . . . . .153 

Miscellaneous formulas 155 

Index 159 



"Perhaps we should all know hozv to use a tool as fine as the 
calculus." — J. McKeen Cattell. 

"A man learns to use the calculus as he learns to use the 
chisel or the file on actual concrete bits of work." — John 
Perry. 



CALCULUS. 

CHAPTER I. 

DIFFERENTIATION AND INTEGRATION. 

1. In the experiences of every-day life and in the 
operations of science, people are continually dealing 
with things which keep changing in quantity, and with 
things so connected that a change in one of them is 
followed by a change in another. For instance, we 
know that work varies in amount, and that the amount 
done by workmen depends on the time. We find crops 
now abundant and now scanty ; and, other things being 
equal, the crops are seen to vary with the amount 
of fertilizer used. We observe that the height of the 
mercury in a thermometer changes with the temperature. 

In these examples and all similar ones there is a 
relation of cause and effect, or at least a relation of 
antecedent and consequent; and Ave say that a quan- 
titative change in the cause is accompanied by a 
quantitative change in the effect. 

It is part of the business of science not only to dis- 
cover relations of cause and effect, but also to try to 
express these relations with precision. When a relation 
of cause and effect can be stated with exactness, the lan- 

1 



2 CALCULUS. 

guage of mathematics is the best one to use, because it 
gives compact and unambiguous expressions, and because 
a further examination of the relation may then be con- 
ducted in that language and results easily reached which 
could be arrived at only with much difficulty, if at all, 
in any ordinary language. For example, the expansive 
force of air was a property observed by Guericke (1602- 
1686), but Boyle (1627-1691) discovered that the vol- 
ume varies inversely as the pressure. That is, if v 
represents the volume of a given quantity of air and p 

its pressure on unit area of the containing vessel, vac-, 

P 
and pv — a constant. We shall see later how we may 

learn more about this law by using the equation pv = c. 
Experience in balancing bodies of equal or unequal 
weights no doubt furnished ancient craftsmen with 
some vague notions regarding equilibrium ; but Archi- 
medes (287(?)-212 B.C.), from a few assumptions, con- 
cluded that two bodies suspended from a bar are in 
equilibrium when their distances from the point of 
support of the bar are inversely proportional to their 
weights. That is, if I, V are the distances of the bodies 
whose weights are w, w', respectively, w : w' : : V : L The 
principle of the lever, as thus stated by Archimedes, 
was later fully established. To illustrate further, from 
earliest times men must have noticed that unsupported 
bodies fall to the ground ; but after the investigations 
made by Newton (1612-1727), it was possible to state 
the law of gravitation with mathematical accuracy : 
The mutual attraction (stress) between any two bodies 
varies directly as the product of their masses and in- 
versely as the square of their distance from each other, 



DIFFERENTIATION AND INTEGRATION. 3 

Thus, if F is the whole attraction between the earth 
and the moon, for instance, M the mass of the earth, m 
the mass of the moon, and r the distance between them, 

F = c — — . These examples go to show that when a 
r 2 

precise quantitative statement can be made in science, 
mathematics, with its unambiguous symbolic shorthand, 
offers the most economical way of making it. 

2. Two modes of quantitative change or variation 
present themselves. As an illustration of the first, the 
number of roses in a handful may be varied by adding 
one and another and another, until the number has 
changed from a to b ; or we may add several at a time 
until the number has changed from a to b. But we 
cannot do less than add one wdiole rose at a time ; for, 
in this case, the variation element is a whole unit, that 
is, a whole rose, and not any fraction of it. Again, we 
may measure a day with a minute as a unit of measure, 
and say that a day contains 1440 minutes ; but this is 
only an artificial convenience. Time does not increase 
a minute " at a time,*' or even a second at a time, but 
by elements of time which are immeasurably small frac- 
tions of a second. This is the second mode of varia- 
tion : a quantitative change not by jumps or finite 
amounts, but by indefinitely small amounts. 

3. By the term variable we mean a quantity which 
changes in the second manner above described. We 
use it in speaking of such things as volume, pressure, 
distance, etc., when they are conceived as being in a 
state of continuous variation. The term function is 
applied to the quantity which necessarily changes be- 



4 CALCULUS. 

cause of a change in a variable with which it is con- 
nected. For example, the pressure of steam on the 
piston of the cylinder is a function of the volume of 
the steam. The attraction which the earth exerts on 
the moon is a function of the distance of the moon 
from the earth. 

If the symbol x stands for the variable and y for the 
function, we briefly express the fact of their connection 
by the general statement y =f(x). The precise nature 
of the connection is shown by specializing /(V). For 
example, if f(x) is log x, we have the particular 
statement y = log x. 

When we need to distinguish one function from 
another, we use such forms as FQx), <$>(x), u, v, etc. 
The nature of these conventional symbols should be 
carefully noticed. The parenthesis merely serves to 
separate the quantity symbol x from the other symbols 
/, F, etc., which are not quantity symbols and hence 
not factors. /(#) is only algebraic shorthand for the 
expression "a function of the varying quantity #." 

Functions are classified as algebraic and transcen- 
dental. An algebraic function is defined as one which 
implies only a finite number of the algebraic opera- 
tions, addition, subtraction, multiplication, division, 
involution, and evolution. 

The trigonometric functions sin#, cos^, tanx, etc., 
are transcendental; so also are e x , log a;, sin" 1 #, etc. 

4. Let us now suppose a change in pressure, or dis- 
tance, or time, or whatever quantity we are dealing 
with under the symbol x. Let Sx stand for the amount 
of change. Then the new quantity is represented by 



DIFFERENTIATION AND INTEGRATION. 5 

x + &r, and f(x) becomes f(x + Bx). Subtracting tlie 
former value from the latter, we have f(x + hx) — f(x), 
the amount of change in the function occasioned by the 
change in the variable. It may be represented by 8y if 

y=f(x). Then ^* ^/ — , or its equal -^, is 

ox ox 

the ratio of the increment of the function to that 
of the variable. Let Sx now be supposed to become 
smaller and smaller until we cannot tell the difference 
between it and zero. We say it " has zero for its limit," 
or it " diminishes without limit," and to show that this 
supposition has been made we use the symbol dx in 
place of 8x, and also use dy in place of 8y for the indefi- 
nitely small change in the function, dx is called the 
differential of #, and dy the differential of y. The ratio 

-^ is called the first differential coefficient oif(x) with 
dx 

respect to x, or briefly, the derivative. 

dxi 
We shall find that the ratio -p is itself, in general, 

some function of x ; hence it is often written f (x). 

The symbols —f(x), -^, f r (x) all mean the same 

thing. It is important to notice that although dy and 

d v 
dx, the individual terms of the ratio -j-, are indefinitely 

small, the ratio itself is usually finite. 

This dx of the mathematician is suggestive of the " atom " of 
the chemist, the " particle " of the physicist, and even the " cell " 
of the biologist. It is the ultimate element of that with which the 
mathematician deals, and always implies one property of the quan- 
tity symbolized by x ; namely, its continuous variation. 



b CALCULUS. 

5. To illustrate the nature of -~ let f (x) — ttx 2 , the 

dx 

area of a circle. Suppose this circular area to be cut 
out of a thin sheet of metal and to have heat supplied 
to it in such a way as to cause it to expand, but to re- 
main circular. Let the radius increase by the amount 
Sx ; then 

y + 8y — f(x -f- 8x) = ir (x + 8x) 2 

= 7TX 2 + 2 TTxSx + IT (8xJ 2 , 

and 8y = f(x + Sx) -f(x) = 2 wxSx + it (8x) 2 ; 

By 
hence # = 2 irx + tt8x = it (2 x + 8x). 

Now, if 8x be made indefinitely small, the limit of 
2 x + 8x is 2 x, and therefore 

^ = 2™. 

This means that if the quantity of heat used is so. 
small that the increase in the length of the radius is 
indefinitely small, the ratio of the increment of the 
area to the increment of the radius is equal to the cir- 
cumference of the circle, a result which might have 
been guessed beforehand if we had reflected that the 
growth in area is a belt only dx wide around the circle, 
and dx is "next to nothing." 

As another illustration, suppose we take the equa- 
tion 2/ = -, which states the law concerning the mutual 
x 

dependence of the volume and pressure of a gas, x repre- 



DIFFERENTIATION AND INTEGRATION. 7 

senting volume and y representing pressure. Let us 
look at pressure as a function of volume ; then 

Ju 

and y + By = f(x + &z) = 



therefore By = 

and 



x + 8x 

c cBx 



x + Bx x x(x + Bx) 

Sy = g__ . 

& x(x + Bx} 



hence 



dy _ 
dx 



In other words, the ratio of the increment of the 
pressure to the increment of the volume is inversely 
as the square of the volume. The minus sign means 
that when the volume takes an increment the incre- 
ment of the pressure is negative ; that is, the pressure- 
increment is really a decrement. This agrees with what 
is said by the equation itself ; namely, that the press- 
ure decreases as the volume increases, and vice versa. 

If the student will follow the thought in a few con- 
crete examples like the two just given, he will gain a 
better insight into the nature and purpose of the cal- 
culus than he can acquire from the mechanical working 
of a great number of meaningless exercises. 

6. The importance of the ratio -^ is soon realized 

dx 

by the student of the mathematical side of any of the 
sciences which admit of mathematical treatment, such 
as astronomy, thermodynamics, electricity, chemistry, 



8 CALCULUS. 

economics, etc. Granting its importance, we should 
know how to find, by the most direct method, the 
derivative of any ordinary algebraic or transcendental 
function ; and, what is even more essential, we should 
be able to perform the reverse operation ; that is, hav- 
ing -f- =z f'(x) to find /(a;). In the following articles, 
ax 

theorems are established by means of which derivatives 
may be directly written, so that we need not take any 
intermediate steps as in Art. 5. 

This operation of finding derivatives is called differ- 
entiation. We begin with a function which is itself 
the sum of two functions. 

7. Let u = some function of x, and v some other func- 
tion of x, and let 

y=f(x)=u + v. 

Then, if x takes the increment &r, 

y + By = f(x + Bx) = u + Bu + v + Si>, 

and By = Su + Sv ; 

, By Bu , Bv 

hence -f- = — + — > 

ox ox ox 

and in the limit -f- = — + — — (1) 

ax ax ax 

It is evident that the same proof applies to any number 
of functions connected by plus and minus signs. 

A constant quantity, because it is a constant or un- 
varying quantity, has no increment ; and if we attempt 
to express its derivative, we have nothing to divide by 
dx. This amounts to saying that the derivative of a 
constant is zero. 



DIFFERENTIATION AND INTEGRATION. 9 

8. Let the given function consist of the product of 
two functions as expressed by 





y=f(x)=uv. 


Then 


y + Sy = f(x + &r) = (it + Sii) (y + hv) 




= uv + vSu + uSv + SuSv ; 


hence 


Sy = vSu + uSv + SuSv ; 


therefore 


Sy Su , Sv , ^ Sv 
-f- = v y- + u — + Su — , 
ox ox ox ox 


and 


dy _ du dv 
dx dx dx 



(2) 

AM UJU 

The last term Su — is disposed of by observing that 

ox 

as Su diminishes without limit, any quantity (except oc) 
multiplied by Su diminishes without limit, and is there- 
fore dropped. 

Similarly, if y = uvw ■•- where u, v, w, etc., are func- 
tions of x, 

dy , ^du . , A dv , . ^dtv , , 

_£ = (w • ••) — + (ww •••)^- + (^ z; -- # )-r--l ^ etc. 

ax ax ax ax 

9. Taking the last expression of the preceding article, 
suppose v = u, w = u, etc., so that uvw ••• becomes (u) n , 
n being the number of functions of x. Then y = u n , 
and the expression 

dy , ^du . r A dv . , N a 1 ^ . , 

-f- = (vw -•0^ + Quw---)— + (uv -..) — + ••, etc., 
ax ax ax ax 

ax ax ' 6?x 



10 CALCULUS. 

This is a polynomial of n terms ; hence we can write, 

(3) 



dy n _ Y du 

-2- = nu n l — 

dx dx 



As a special case, if 

u = x, v = x, iv = x, etc., 
y = x n , 
and formula (3) becomes 

— £ = nx n ~ l . 
dx 

p 
10. Let y =f(x)== u q in which p and q are constant 

quantities, positive and integral. 



Then 


y q =u*, 


and by (3), 


„ i dy n ,du 


hence 


dy p u p ~ 1 du 
dx q y q ~ l dx 



Eliminating y from this expression by means of the 

V 

given expression y = u q , we have 

dx q dx 

11. Let y =f(x) = u~ n ,n being integral and positive. 

Then y = — , and so yu n = 1. Using the formula for 

u n 



DIFFERENTIATION AND INTEGRATION. 11 

the derivative of the product of two functions, and 
writing zero for the derivative of unity, 

U n _M_ _j_ y nu n-\ __ _ Q . 

dx dx 

,, , ■ dy ynu n ~ l du nu n ~ l du 

that is, -f- = — ^ _ = _, 

dx u n ax u m dx 

or -f- = — m^ w x — (5) 



12. Comparing formulas (3), (4), (5), it is seen that 
if y — u n , in which w is a function of x and % is any 
constant, 

dy „_i c?m 

cfe dx 

The translation of this formula gives, therefore, the 
only rule that is needed for finding the derivative of 
a function affected with any constant exponent. It 
should be noticed, however, that since this expression 

cl ii all f 

for the first derivative, -^-, contains — as a factor, we 

dx dx 

may require various other rules if we are to find the 

expression for which — is the symbol. For example, 

suppose y — (log #) 3 . We now know that -^- = 3 (log x) 2 

multiplied by the derivative of log x, whatever it is. 
What it is we shall learn in a subsequent article. At 
present we can only write 

-Jl = 3 (log x) 2 — log x. 
dx dx 



12 CALCULUS. 

13. For the ratio of the indefinitely small increment 
of the function to the increment of the variable we may 

of course use — f(x), as well as the symbol -^. It 
dx dx 

should be carefully noticed that the d standing above 

dx is here, as everywhere, a symbol of operation and not 

of quantity, signifying the differential oif(x). Notice 

also that an indicated operation counts for the same as a 

sy£i ry*& 

performed one. For example, in an operation 

a ~\~ x 

/y* /y*a 

is indicated, and the expression has everywhere 

the same value as a — x, the result obtained by actually 

d 
performing the division. So, for example, — irx 2 has 

the same value as 2 ttx. 



14. Differentiation is seen to be a tearing down 
process, whereby we reach an ultimate element of quan- 
tity. The reverse operation, one of building up, is 

known as integration; its symbol is ( (long s). 

As already shown, rules are established for the 
differentiation of functions, but integration is largely 
a matter of guesswork and experiment. The test of 
the correctness of any integration is this : differentiate 
the result; we should get the given differential form. 

Tables of integrals enable the student to write 
directly the expression corresponding to an indicated 
integration, so that he need not go through the process 
of discovering the required expression. 

In accordance with what has just been said, we have 

J dy = y, the symbols 6?, J , neutralizing each other. 





DIFFERENTIATION AND INTEGRATION. 


Also, 


,if 


dy _ du dv 
dx dx dx 

dy = du + (7y + ..., 


and 




# = w + v + .... 


If 




dy _ cfe c/^ 
e?# dx dx 



13 



y — \ (ydu + wtfo;) 



tw. 



If 






# 



m + 1 

15. The symbol \f(x)dx is known as a general or 

indefinite integral. After discovering a function, say 
(/>(#), which differentiated will give f(x), we ought to 
write 



ff(x)dx = cf>(x)+C, 



in which (7 is a quantity primarily undetermined, and 
known as the constant of integration. Since C is a 
constant, 

-f O (x) + €T\ = ±+ (x) + -f (7=/<», 

and as a constant term may thus exist in connection 
with the original function we give the integral the 
benefit of the doubt and write as stated, cf> (x) + 0. 



14 CALCULUS. 

16. The symbol I f(x)dx is known as a definite 

integral. Its meaning is this : Find the general inte- 
gral, which will be some function of #, and substitute b 
for x ; then substitute a for #, and subtract the latter 
expression from the former. To state the process sym- 
bolically, let 

)f(x)dx = 4>(x)\ 



j> 



then I f(x)dx = 4>(x) 



= £(*)- <K«> 



a and b are called the limits of the integral. The 
constant O disappears since 

[>(&) + C]-[<k«) + C] = </>(£)- <K«). 

For further discussion of definite integrals, see Art. 73, 
and for other methods of finding the value of (7, see 
Arts. 60, 61, etc. 

Exercises. 

17. 1. If y = nu, -¥- = n — . Prove this in two ways: 
dx dx 

(1) by beginning y + 8y = n (u + 8u) ; (2) by regarding nu 
as a special case of the product of two functions in which 
one of the functions is a constant (a function by courtesy). 

2. If dy = nf(x)dx, show that 

y = J nf(x) dx = n I f(x) dx ; 

that is, a constant factor under the integral sign may be 
removed and written as a coefficient of the integral. Notice 

that if dy = nf(x) dx, -1= f( x ) dx. 
n w 



DIFFERENTIATION AND INTEGRATION. 15 



u 

3. If y = -, show that 

J v' 

die do 

V u 

dy __ dx dx 

dx ~~ v 2 

Translate this formula into a theorem. 

4. Prove that if y = — , — ~ = , m being a constant. 

m dx m dx 

Do this in two ways: (1) by observing that the result in 

exercise 1 holds when w is a fraction ; (2) by making — a 
special case under exercise 3. 

5. Prove that if y = — , 

du 

m — 

dy dx 

dx u 2 

6. Prove that the derivative of the square root of any 
function is the derivative of the function divided by twice 
the square root of the function. That is, if y = Vw, 

du 
dy dx 



dx 2 V 



a 



f(x) dx = — f f(x) dx ; that is, the limits 
may be reversed if the sign of the integral is changed. 

8. If y is the surface of a sphere whose radius is x, • 

dx 

9. Show that the ratio of the differential of the circum- 
ference of a circle is to the differential of its radius as 2 ir : 1. 



16 CALCULUS. 

18. Thus far y has been in each instance an explicit 
function of x. Suppose now that x and y are so com- 
bined in any expression that y is only implicitly a 
function of x ; for example, as in the expression 
ax + by + c = 0. This statement is the equivalent of 

the explicit statement y = . A little algebraic 

consideration will show that it is unnecessary to solve 

for y before proceeding to find — ^. We may differ- 
ed 7 

entiate immediately, and then solve for -f-. Thus, if 

dx 

ax + by + e = 0, differentiating term by term with x as 

the fundamental variable, we have a + b -^ = ; and 
7 dx 

hence -f- = — -, which is the same result that we get 
dx b _ 

by differentiating y = 



Exercises. 



19 



l.£ + £=l 5 showthat^=±^. 



a 2 b 2 dx a -y/tf 

Differentiating immediately, 

2 x dx . 

-T+-T2— =°; 



dy dy __ b 2 x 

dx dx a 2 y 



solving for -f->, 



y may now be eliminated by using its value 
b 



± - -ya 2 — x 2 , 
a 

obtained from the given equation. 





DIFFERENTIA TION 


AND 


INTEGRATION, 


2. 


# 8 + y 2, - 


- 3 axy = ; fin 


dx 






3. 


x s + y 3 - 


- 3 axy + a 3 = 


; find 


dy 
dx 




4. 


pv = c ; 


show that -J- = 
dv 


c -p 

: - - if J) = 


= /(*)■ 


By 


formula 


,(2), *f +i > = 


'•> 




hence 


'J 


dp _ 

dv 


i ; = - 


G 

~V 2 ' 





17 



5. 27v k = c'] show that — = -• 

Here, and also in exercise 4 ? it is just as well to solve 
for p before beginning to differentiate. We have p = — , so 
that vk 

dp _ _ kc'v k ~ l _ _ kc' 

dv v 2k v k+1 

6. F= c — — ? in which c, M, m are constants ; show that 

c]F = _2GMm m 

dr r 3 

It is often desirable to use other letters besides x and y to denote 
the variable quantities. The student should therefore accustom him- 
self at the outset to such symbols as those given in exercises 4, 5, 6. 

20. If f(x) is a varying angle, it is clear that any 
trigonometric function of the angle must also vary. 

We have now to find -j- , when y represents each one 

of the trigonometric functions in turn. 

Suppose y = sin/(^). Let f(x) = u ; then y = sin u, 

1 Sy _ sin (u + S-u) — sin u 

(XllKX — — ~ 

OX ox 



18 CALCULUS. 

Put u + Su = a, and u = {3 ; 

then, since sin a — sin /3 = 2 sin J (a — £) cos J (cc + /3), 

Sy = sin (w + Su') — sin w = sin a — sin /3 

= 2 sin | Sw cos \(2>u + hu) 

= 2 cos (u + I Sw) sin 1 cm ; 

, S?/ ., ^ N sin ^- Sw S^ 

hence ^- = cos (u + A- ou) — j-g — • -k— 

ox A J \ou Sx 

But when an angle diminishes without limit, we may 
write the angle itself for its sine ; so we now have 

dy d . du sn\ 

-~= — sin u = cos u — — (b) 

dx dx dx 

Let y = cos w = sin ( — — u J ; 

,i dy d . (it \ (it \ d (tt \ 

then -^ = — sm — — u )= cos — — u) — — — u ; 
^ ^ V2 y V2 Jdx\2 J 9 

. i n ay a • au ^tn 

thereiore -^- = —oosu= — smu—— (7) 

\AiJU \XlJU \AlJU 

sinw 

Let V = tan u = 

cosw 

Using exercise 3, Art. 17, together with formulas 
(6) and (7), 

dy cos 2 u + sin 2 u du 9 du 

J = 2 1~ = SeC M TT 5 

dx cos A u ax ax 

therefore -M. = — tan u = sec 2 u — • (8) 



BIFFEBENTIATION AND INTEGRATION. 19 

T . COS u 

Let y = cot u = — 

sin u 

du — sin 2 u — cos 2 u du 9 du 



(9) 



dx 




sin 2 u dx 




therefore 




dy d 

-£- = — cot u = — 

\XiJu LI il/ 


9 C?7£ 

cosec" 5 m — 
dx 


Let 




1 

y = sec u = : 

u COS 76 




then, using 


exercise 5, Art. 17, 






dy 


du 

sin u — 
dx 
= = = tan ?/ ser 


du 

*, u — % 



and Ave have 



dy d , du /1AX 

-^- = -—sec 7^= tan u secu — (lu) 

dx dx dx 



Let y = cosec u 



SUlTi 

— cos u — 
dy dx t du 

-^- = — } = — cot u cosec u -^-, 

ax sur u dx 

and therefore 

dy d , du .---,. 

-f- = — cosec u= — cot u cosec w — (11) 

21. In formulas (6) to (11) inclusive we have the 
ratio of the differential of the trigonometric function to 
the differential of the angle x. These formulas, which 
are remarkable for their simplicity, should be translated 
and committed to memory. We may next regard the 
trigonometric function as varying by equal increments, 
and thereby causing a change in the angle. From this 



20 CALCULUS. 

point of view we have to find the ratio of the differen- 
tial of the angle to the differential of the trigonometric 
function. 

22. Let y = sin -1 u \ then u = siny. 
Hence, by formula (6), 

du dy 

— = cos y -j- ; 
dx ° dx 

du 

dy dx 



that is, 



dx cos y 



But cos y = Vl — u 2 ; 

du 

therefore §y _ —^ n -i u = ^ . (12) 

cfa cte Vl-i( 2 

Let 2/ = cos -1 1£ ; then w = cos y, and proceeding as 

before, Ave find 

du 

dy d - tfe ^-, ox 

-f- = -—cos" 1 ^= — =— (lo) 

a# a# VI _ ^2 

Let 2/ — tan -1 u ; then m = tan y, 

6?W 



6?w dy ,, , . dy dx 

— = sec z ^ -~ ; that is, -^ = — 5- ■ 
a^ a^ a# sec z 2/ ' 

du 



and -^ = sec 2 y -~^ ; that is, -^ 



therefore ^ = _^ tan -i M = _^ ; (14) 

dx ax 1 + w 

similarly, if y= cot" 1 2/, 

dy d ^ . dx sh r x 

-# = -j- cot -1 w = - q 5- (15; 

dx dx 1 + ?r 



DIFFERENTIATION AND INTEGRATION. 21 



Let y = sec 2 u. u = sec y, 



, du dy 

and -7- = tan y sec v -7 1 : 

dx * * dx 

du 

that is, % = dx 

dx tan y secy 



Bnt since u = sec y, tan y = V^ 2 — 1 ; and we have 

du 

dy d ., dfe ^i^n 

-f =-— sec- 1 ^^ , (Id) 

dx dx U -y/y? _ l 

Finally, if y = cosec -1 % we find that 

du 
dy d _- 6fe 



= ^- cosec * % 



dx dx u V^ 2 — 1 



(17) 



All these operations of differentiating may be reversed, 
so that if 

dy du 



dx 



du r , . , 

cos u -j-, y = I cos udu = sm u, etc. 



23. It remains to find -!f- when 2/ = e x , a x , log #, log 1 w, 

e w , a* u being, as before, equal to f(x). 
From algebra* we have 

[2 [3 |4 |n_ 

in which g is the number 2.7182818284 . . . forming the 
base of the Napierian system of logarithms. If y = e x , 

c li = A e * = A_(i + x + ^ + ^ + 

dx dx dx\ 1 2 [3 

*See Hall and Knight's Elementary Algebra, Art. 537. Edition of 



22 CALCULUS. 

Performing the operation indicated, that is, differen- 
tiating this series term by term, 

but this result is the original series which e x equals ; 
therefore 

*£ =*-«*=«• (18) 

dx dx 

This result is unique, being the only case known in 
which the derivative of a function is the function itself. 
We have also from algebra 

a x =1 +xlog e a+ v + 7/ +•••; 

therefore 



— a x = log e a 
dx 



1 + # log e a + - v ,^ e 



# 2 (log e a) 2 
= ^ log e a 



Hence, if y = a x , 



-^- = —-a x = a x log*, a. (1 9) 



24. Let ^/ = log e x ; then # = e y ; and if we regard x 
as a function of y, we have by formula (18), 

dx t . , . dy 1 1 

-=- =e y ; that is, -f-.= — = -; 

dy ■ ax e y x 

therefore -y- = t"1°^ » = — (20) 

dx dx ° x 



DIFFERENTIATION AND INTEGRATION. 23 

dy 
To find -j- when y = log c /(x) = \og e u, we notice that 

«F = ^r- • -s- and in the limit 

OX 0^ OX 

6?^/ dy du 
dx du dx 

Now if y — log e u, -=^ = -, by formula (20), 

CtU U 

and therefore -^- = -—\og e u = — =-• (21) 

dx ax u dx y 

If y = e u , log e y = u 

and -r= - = -r- log y = - -¥-, by formula (21), 

dx dx ** y dx J v J " 

. 6?V 6?M 6?W 

hence ^ = y — = <?« -^- 5 

ax ax ax 



,., , dy d du 

so that we have -^- = — - e 11 = e ll -j-. 

dx dx dx 

Finally, if y = a u , log y = u log a, and 



(22) 



du Id 1 dy . 

dx log a dx ^ ^ y log a c?x 

, dy du , du , 

hence ~r = y -=r~ lose # = a M -^- log a, 

c/x ^ dx h dx b 

and therefore -^ = -=-a M = a w - T - log a. (23) 

ax dx dx 

Formulas (1) to (23), together with the correspond- 
ing integration formulas, are collected in Chapter V. 



24 CALCULUS. 

25. Since first derivatives are themselves generally 
functions of the fundamental variable, their first deriva- 
tives may in turn be found. These last are called second 
derivatives of the original functions. Second deriva- 
tives are indicated by the symbols f n (x), — ( — )' — o- 

dx\dxj dx 2 

It is to be carefully noticed that " <i 2 ," like d, is not a 
symbol of quantity, but of operation. The d 2 should 
never be read u d square," but " second d" Since 

— - always means — \^\ it is best to use the latter 
dx 2 dx\dxj 

form until there is no danger of misunderstanding the 

d u 
symbol -~^ The derivative of the second derivative is 

(XX 70 

d V 
called the third derivative and is written -~; the next 

<Py 

is written — -> and so on. 
ax* 



26. In what has preceded we have assumed one 
fundamental variable ; but reference to common ex- 
amples shows us that we may have a function of two or 
more independent variables. For instance, crops vary 
not only with the amount of fertilizer used, but also with 
the amount of sunshine and moisture. 

If z is a function of two independent variables x and 
y, expressed by writing u = f(x, y), we may differen- 
tiate, supposing x to vary and y to remain constant, or 
Ave may suppose y to vary and x to remain constant. 

In the former case we have — -; and in the latter, — . 

dx dy 

These ratios are known as partial differential coefficients, 

and to indicate this we may use the parenthesis, writing 



DIFFERENTIATION AND INTEGRATION. 25 

( — j and t — V When we suppose x and y to vary 

simultaneously the corresponding change in the func- 
tion is called the total differential. As an example 
of partial differentials suppose we have pv = nt, where 
p and v are pressure and volume as before and t is 
the absolute temperature of a gas. Suppose t varies 

while v remains constant ; then f-^ J == — Again, let v 
vary while t remains constant, I -J-] — . — — , as in exer- 

a a j. -tr\ \dvj V A 

cise 4, Art. 19. x y 

27. Having z = f(x, if), let us differentiate z with 
respect to x and then differentiate the result with respect 
to y. The order of the steps is indicated by 

d fdz\ d 2 z 

■ — — or 

dy \dxj dydx 

The reverse order is indicated by 

d f dz\ d 2 z 

■ — — or 

dx\dyj dxdy 

We proceed to show that 

d f dz\ __ d f dz\ . 
dy \dx) dx \dyj 

that is, we get the same result in whichever order we 
proceed. 

Let x take the increment Sx while y remains constant ; 

then — = /<> + g ^?/) -/(^y) , 

Sx Sx 



26 CALCULUS. 

Now let y in this result take the increment Sy while x 
remains constant. 

By \8xJ 

= f(x + &e, y + 8y ) -/(a?, y + 8y) -f(x + 8a;, y) +/(>, y) 

SySa; 

Reversing the order, we have 

8z = f(x, y + 8y) -/(a;, y) 
8y 8y 

8_(8z y 

8x\8yj 

_ f(x + 8x,y + 8y) -f(x + &k, y ) -/(a;, y + Sy ) +/(s, y ) ^ 

SxSy 

Hence _—=—(—; 

oy\oxj cx\oyJ 

and in the limit 

» 

d fdz\ _ d fdz\ m d 2 z _ _ d 2 z 

dy\dxj dx\dyj ' dydx dxdy 

In any scientific investigation in which the calculus 
is used the context must show what and how many 
variables are involved, and what partial differential 
coefficients will occur. For example, Carnot's Prin- 
ciple, with its applications as presented in thermo- 
dynamics, affords an abundance of cases of these partial 
differential coefficients. 

28. Ordinary text-books in algebra and trigonometry 
usually give methods for expanding (a + x) m , e x , a x , 
log(l + x), sin a;, etc., into series in ascending powers 



DIFFERENTIATION AND INTEGRATION. 27 

of x. We can now establish one general theorem by 
means of which these functions and all similar ones 
may be expanded. 

We first notice that if y =f(z + x) and we differ- 
entiate regarding x as a variable and z as a constant, 
the result is just the same as if we should differentiate 
with z for the variable and x as a constant. That is, if 

y =f(z + x), (-¥- )= ( — )• This is obviously true if 

we consider that it makes no difference whether we 
change the function by changing z or x. 
Suppose 

f(z + x)= A + Bx + Ox 9 - + Dx* + E& + ..., (a) 

in which A, B, C, etc., are functions of z and not of x. 
Let us now differentiate successively the first member 
of equation (a) with respect to g, and the second mem- 
ber with respect to #, and put x = after each differ- 
entiation. 

Then, since — f(z + x) = — f(z + x), 
dz ax 

dz 

= B + 2Cx + 3Dx 2 + 4:I]x 3 + ..., 

and f(z)=B. 

f"(z + x) = 2 + 2 • 3 Dx + 3 • 4 fix 2 + ..., 

and /"(s)=2<7. 

f"(z + x)=2-SD + 2-3-4Hx+ -, 

and /"(*) = 2 -8 2>. 



28 CALCULUS. 

Also, if x = in equation (a), f(z) = A. 
We now have 

A =/<», B=fXz), 0=\f"(z), etc. 
Putting these values into the assumed series (V), 
/<> + aO=/<V>+/'(3> + -^> 2 + /^! + 

[n 

This formula is Taylor's theorem. It enables us to 
expand functions of the sum of two variables in ascend- 
ing powers of one of the variables, combined with finite 
coefficients depending on the other variable. 

29. Suppose we have a function of one variable and 
wish to expand it into a series. Following the method 
of the preceding article, assume 

f(x) = A + Bx + Ox 2 + Dx 3 + Eat ■ •-. (5) 

Differentiating successively and putting x = after 
each differentiation, we have 

f(x) = B + 2Cx + 2,Bx i + 4Bx z + ..., 

and f(*)\ = B. 

f"(x)= 2C + 2 ■ 3 Dx + 3 ■ 4 Bx 2 + ...,. 

and /"<»] o = 2a 

f'"(x)= 2-3D + 2-3-4& + -, 

/'"(z)] o = 2.3D. 



DIFFERENTIATION AND INTEGRATION. 29 

Also, f{x-)\ = A. 

The assumed coefficients A, B^ etc., are thus deter- 
mined, for we have 

A =/(V)] , or, as it is usually written, /(0); 
£=/'(*)]o=/'(0); 

"" ~ [3 " [3 ' 

Putting these values into the assumed series (5), 

/'(0)z 2 /'"(O)s 8 



f(x)=f(P)+f(0-)x + ^-^ + "—^ 



- + 



, j^CQ)^ , 



This formula is Maclaurin's theorem. 

It will be observed that if z is made equal to zero in 
Taylor's theorem, we have Maclaurin's theorem. The 
latter may therefore be regarded as a special case under 
Taylor's theorem. 

30. Suppose f(x) = (a-\-x) m \ 

let us expand this function according to Maclaurin's 
theorem. 

If x = 0, the function becomes a m . 



30 CALCULUS. 

Further, /' (x) = rn(a + x) m ~\ 
and /'(0) = m O + ^) m_1 ] = ma m ~ l . 

f n (x) =m(m — I) (a + x) m ~ 2 , 
f" (0) = m (m - 1) (a + ^) w ~ 2 ] = m (m - 1) a™" 2 . 

f"(x)=m(m — l)(a + ^) m " 3 , 
and hence 

/'"CO) = m(m - l)(w - 2)(a + x) m - 3 ] 

= m (m — 1) (m — 2) a m ~ 3 . 



Therefore Ave have 
(a + ^) m = a m + ma m ~ l x 4 



m Cm — 1) a m 2 # 2 



[2 



m(m— 1) Cm — 2) a™ V 



m(m - 1) - (m -?i + 2)a w - (w -V- 1 
|w — 1 

This formula will be recognized as the binomial 
theorem. It provides for the expansion of a binomial 
affected with any constant exponent. 

31. A series must be known to be a converging series 
before any practical use can be made of it. The sim- 
plest tests for convergency are given in algebra text- 
books.* If a series is found to be diverging, it is 

* See Hall and Knight's Elementary Algebra, Arts. 470-477. 



DIFFERENTIATION AND INTEGRATION. 31 

rejected for such values of the variable as make it 
diverging; or it is transformed into a series which 
converges and, if possible, into one which converges 
rapidly, in order that only a few terms need be used. 

32. If one function is divided by another, as *. A 

it sometimes happens that the functions are of such 
a nature that upon evaluating them for some particu- 
lar quantity each function reduces to zero, so that we 

have -• The question arises : what does this expres- 
sion mean, and what is its value ? 

Students often say that - must be unity ; sometimes 

they are inclined to think it is zero. In some instances 

the first view is correct ; in others, the second ; in others 

still, a value will be found which is neither unity nor 

zero. Now it is evident that if we can find a limit 

f(x) 
which the ratio \) J is approaching as x approaches 

nearer and nearer to that value which makes f(x) and 
<j>(x) each equal to zero, we have caught the correct 

value of — We proceed to find a general expression 

for this limit. 

Suppose x to take the increment Bx ; then by Taylor's 
theorem, 



32 CALCULUS. 

Let a be the quantity that makes both f(x) and <f>(x) 
equal to zero. Substituting a for x, 

/0, + fa) HA.)i.^W t ... 

rv J + <£'<>)&* + r ^ y (&e) 2 + ... 

Dividing both numerator and denominator of the 
second member of this expression by 8x, 



Finally, when &r becomes c?:z, 
f(a) = f'(a) 

/"<» 

Hence, if ^ becomes x when evaluated for any 

quantity as a, the value of this indeterminate form is 



<£'o.) 



evaluated for a. 



If it should happen that ^j 



= -, we divide both 
<f> f (x)\ 

numerator and denominator of equation (2) by Sx again 

and have 

/O) f"(a) 



</>(a) <£"<» 



/CO 



If , ) ; becomes ^ when evaluated for any quantity 



DIFFERENTIATION AND INTEGRATION. 33 

as a, this expression may be determined as in the first 
case by observing that if 



/O) = oo <Kx) _ o 
4>(x) °°' l o" 

If f(x)<j>(x') = oo . when evaluated for some quantity, 
<f>(x) 
— T~~~ = <? wn ^ cn ma y ^ e treated as above. 

Too 

If f(x)— <£(#) = oo —go for some quantity, it should 

be transformed into a fraction which takes the form - 
and then determined. 



33. If y = \_f(x)^ x \ log y = <j>(z)logf(z); and this 
is indeterminate whenever one of the factors becomes 
zero and the other infinite for the same value of x. 

(1) Suppose $(x)= 0, and \ogf(x) = ± oo ; then 
f(x) = cc or 0. Consequently [/(V)]^ becomes inde- 
terminate when for some value of x it takes the form 
0° or oo°. 

(2) Suppose <£(V)=±ao, and logf(x)= ; then 
f(x) = l, and \_f(x)~Y ){x) gives the indeterminate forms 
1" and 1~°°. 

Hence, if we have any of the indeterminate forms 0°, 
oo°, 1~°°, as the result of evaluating [/(V)]^ f° r some 
quantity, we change the exponential function to the 
corresponding logarithmic function, and then reduce to 

the form -, which is dealt with under the first case. 



34 CALCULUS. 



Exercises. 



34. 1. Show that — cotit= — cosec 2 ^ — through the 

iXJu (XX 



relation cot u 



= tan \- — ii\ 

v2 ; 



2. If y = [/(a)]* (x) = v?, show that 

-JL = VU V 1 1- (log U) U v 

dx dx dx 

Take the logarithmic form, logy=vlogu, and differentiate. 

3. y = x x ; -^ = of (1 + log x). 

dx 

-- dy 1 

4. y = e x ; -^ = 

dx - 

a-V 

6. N= e\ tan ,P — log tan (45° + \ F), in which the varia- 
bles are N and F. Show that 

c^Y^ A^-cosi^ 7 ) 
dF cos 2 F 

— Watson's Theoretical Astronomy, p. 69. 

7. Verify the following expansions by means of Mac- 
laurin's theorem : 

/y»£ /y»0 /Vl4 /yffl 

(ii) a^ = l + a! log e a + ^^ + ^f^- 3 +"-. 
(in)Iog(l + a») = »-| + f-J+-(-ir 1 f-. 



DIFFERENTIATION AND INTEGRATION. 



35 



/y»3 /v»5 

(v) C psa! = l-- + - . 

Show that (v) might be derived directly from (iv), since 



— sin x = cos x. 
dx 


8. Show that • = 1. 

X 

-•o 

In this case f(x) = cos x and <£'(V) = 1 ; ^L? 

_ 


9. Show that 

X 


-log- 



= 1. 



10. Show that x l°g( 1+a; ) " 
1 — cos as 



= 2. 



We have 



/ ^)_ l0 ^ 1 + ^) + ^(iT^) 



*'(*) 



sma; 



. 



but this expression evaluated for is - as before. Hence 
we proceed to the second derivatives and have 



■ + ^— \-x 



1 



<£"(#) cos# 

and this equals 2 when evaluated for 0. 



11. Show that 



1 — sin x — 2 sin 2 # 
1 — 3 sin x -\- 2 sin 2 x 



= 3 when a? = 30°. 



36 CALCULUS. 

12. Expand sin -1 a;. Using Maclanrin's theorem, 
fix) = sin" 1 x = /(0) + /'(0) x + .M x* + tUlp. x 3 + - 

/(O) - 0; /'(*) = -J=J /'(O) = 1. 

VI — x- 

f"(x) = ^—;f'(0) = 0. 



_ (1 - a 2 )* - -I x 

ii-xy _ 



Therefore sin -1 sc = a; + 



= 1. 



13. Show that 2r sin" 1 — = a(l+-^- +...). 

2r V 24r 2 ; 

— Thomson and Tait's Nat. Phil., Vol. 1, Art. 131. 

Substituting — for x in the expansion of sin -1 a?, and 
2 r 

multiplying by 2 r, we have the result 

1 + — + 
24 r 2 



14. Expand sin 2 - to the second power term inclusive. 

2 02 

Ans. -j' 

15. Given lajlogojeZa?; perform the operation indicated. 

We have d(uv) = udv + vdu ; (formula 2) 

hence, m; = I \idv + j i>cfa^ (Art. 14) 

or, I udv — uv — I i>(^. 



DIFFERENTIATION AND INTEGRATION. 37 

In the present case let 

u = log x ; then dv = xdx, and du — -- 

x 

X X 

Integrating dv, v = — , and uv = — log x ; 

therefore ( x log xdx = — log x — ( — — 
J 2 J 2 x 

— — \ogx 

2 & 4 

This process is known as integration by parts. 

16. Use the -method of example 15 in the following 
examples : 

(i) I x cos x dx = x sin x + cos x. 

(ii) I e ax x dx = — (x ] • 

(iii) I sin -1 x dx = x sin -1 x + Vl — x 2 - 

(iv) I log x dx = x log x — x. 

17. Given the following integrals, to find their values: 

(i) I (a — x) n dx. 

(ii) I t&nxdx. 
Let cos x,= u; then du = — sin x dx, and 

/tan xdx = | dx = — | — = — log u = — log cos x. 
J cosx J u 



38 CALCULUS. 

(iii) I sin x cos x dx. 
This may be written I i sin 2 x dx, which equals —J cos 2x. 

(iv) I x Va 2 — x? dx. 

Let u == V a 2 — or ; then I a; Va 2 — a; 2 da; = — | m 2 dw 

= _^ = _(a 2 -ar°)j 
3 3 

(v) I Va 2 — a; 2 c?x. 
Let a; = ct sin u ; then I Va 2 — a; 2 da; = a 2 I cos 2 ^dw 
= - f (1 + cos 2 u) cfa = -(it + i sin 2 u) 



= — sin x - H — - V a- — or = — sm A - + - Vcr — x . 
2 V a a 2 7 2 a 2 

(vi) I da;. 

This may be written 

J a 2 __ ( a 2 _ #2\ 
v ' dXj which equals 
Va 2 — x 2 

— dx — I Va 2 — x?dx. 

Va 2 — a? 2 ^ 

Therefore 

^2^" a ^2 " a ' 2 



/Oy t o • _i a; I eh • i x , x / 9 i 

— dx = a 2 sin * — (-sin L - + - Va- — a; 

Va 2 - 



DIFFERENTIATION AND INTEGRATION. 39 

dx 



(vi) /. 



Va 2 + x 2 
Let u — x = Va 2 + x 2 ; 

then I — x = I — - = log i£ = log (a; + Va 2 -f x 2 ). 
J ^tf + x 2 ^ u 

(viii) I Va 2 + x 2 dx. 



Integrate by parts, letting u = Va 2 + x 2 ; 



X LLJu 



then dv = dx* v = a;, du = — , wu = x Va 2 + # 2 , 

and the formula I wcfa; = uv — I v^w 

becomes 

J Va 2 + # 2 dx = x -Va 2 + x 2 — I - 



Va 2 + a 8 



cto 



: a; Va 2 + x 2 — I - 



-dx 



Va 2 + or 5 

= x Va 2 + x 2 - fs/a 2 + x 2 dx+ f 

J J Va 2 + x 2 

Transposing the middle term and dividing by 2, 
I Va 2 + x*dx = - V<x 2 + x 2 + — I 



dx. 



dx 



Va 2 + x 2 



= |V^T^+| log (a? + V^?T^ 2 ). 

35. It should be remembered that the system of 
logarithms used in this chapter is the Napierian. 
Whenever differentiation or integration gives rise to 



40 CALCULUS. 

an expression in which a logarithmic factor occurs, the 
equation containing this factor must be multiplied 
through by 0.43429448 •••, the modulus of the common 
system, before it can be used in computations involving 
common logarithms. For example, in developing the 
theory for determining the place of a comet moving in 
a hyperbolic orbit we encounter the equation 



k~\/p dt = a 2 tan yjr 



l«KH 



in which t and a are the variables. This is to be in- 
tegrated between the limits T and t ; so we have 

*VpJ[*-/rftant[l.(l + l)-g4r 

= a 2 tan^r C-e(\ +~\da -■ C-da ; 

hence JcVp (T— t) = a 2 tan yfr\ -^e(cr j — log e <r . 

Common logarithms cannot be used in this equation 
or in any modified form of it without first intro- 
ducing the modulus of the common system as a factor 
throughout. 



CHAPTER II. 

THE GRAPH. 

36. The changes in a function corresponding to 
changes in its variable may be graphically shown in 
the following way : 

Draw a horizontal line with another line at right 
angles to it. Call the horizontal line XX ! or the 
x-axis ; the vertical line YY' or the y-axis ; and their 
point of intersection or the origin. 

If y is some specified function of x, give a number of 
convenient values to x and find the accompanying values 
for y. Beginning at as the zero point, lay off with 
any convenient unit of length the positive values of x 
to the right on the a>axis, and the negative values to 
the left on this axis. At the end, remote from 0, of 
this line, which represents a value of x, draw a perpen- 
dicular (using the same unit of length) to represent 
the corresponding value of y. The perpendicular is 
to be drawn upward from the #-axis in case y is posi- 
tive, and downward when y is negative. 

In this way locate a point for each pair of values of 
x and y. If many values be given to x, — any two con- 
secutive values differing but little from each other, — 
we shall have a correspondingly large number of points 
with small distances separating them. Connecting all 
the points in order, we have a continuous line, straight 
or curved, called a graph or locus. The values of x are 

41 



42 



CALCULUS, 



called abscissas, and the values of y ordinates. The 
two together are known as coordinates. 

To illustrate, suppose y = x + 2. When # = 0, y = 2 ; 
when x = 1, y = 3 \ when a; = — 2, ?/ = ; etc. When 
# = 0, Ave have no distance to measure off on the #-axis, 
and since y = + 2 we measure upward two units, thus 
locating the point P v Measuring one unit to the 
right and three upward, Ave have the point P 2 . Lo- 
cating a number of points in this, way and then con- 
necting them, the result looks like a straight line. At 
any rate Ave have not been able to get any apparent 
bends or corners — provided the plotting has been 
accurately done. This line presents to the eye the 
way y changes as x changes when y = x + 2. The 
vertical lines representing the values of y seem to 
get steadily longer as x increases. 



X' 



X 



Y 



Fig. 1. 



As another illustration, let us take the isotherm 

c 
equation y = ~ (see Art. 5). Suppose c is unity, so 

1 X 
that y = — When x = 1, y = 1 ; when x = 2, y=\\ 

x 

when x = £, y = 2 ; etc. Locating these points and 



THE GRAPH. 43 

drawing a smooth curve through them, the graph 
appears as in Fig. 1. Two things are clear in regard 
to this graph: (1) it is related to the ^/-axis precisely 
as it is to the #-axis ; (2) as x increases without limit, 
y diminishes without limit, so that the points are nearer 
and nearer to the a;-axis. The graph therefore shows 
what the equation says ; namely, that as the volume 
becomes indefinitely great the pressure becomes indefi- 
nitely small; and conversely, if the volume could be 
diminished without limit, the pressure would be indefi- 
nitely great. 

We further observe that when x is negative, y is negative ; and 
thus the complete graph includes a branch in the diagonally oppo- 
site corner X'OY' (Art. 123). But this second branch represents 
no actual pressures and volumes, because pressures and volumes 
are positive. We shall find numerous instances of equations in 
which the variables, abstractly viewed, have a wider range of 
values than the values possible for the concrete quantities under 
consideration. 

37. If we like, we may think of a graph as the path 
of a looint which moves from one determined point to 
the next one, and thence to the next one in order. 
The equation y = x + 2, for example, merely says that 
the point moves so that its ordinate is all the time equal 
to its abscissa increased by the constant 2. When 
the graph is thus looked upon as the path of a moving 
point, the variable coordinates x and y are called cur- 
rent coordinates. Any equation in two variables may be 
said to express the laiv of the point's motion in the plane. 

For brevity we shall speak of "the curve y=f(x)" 
instead of saying " the curve which the equation 
y=f(x~) represents." 



44 



CALCULUS. 




Fig. 2. 



38. Suppose the moving point describes the arc CO 1 
of the graph or curve y =f(x). Let P be any point 

in the path and Q 
another point. As 
the moving point 
goes from P to Q, 
its abscissa changes 
from x to x + 8x, 
and its ordinate 
from y to y + S?/. 
Draw PL paral- 
lel to the ;r-axis. 
Then PL = &£ and 
LQ = 8y. Let ZT' 
be the chord (produced) passing through the points 

P, Q. —jy is the tangent of the angle which the line 

TT 1 makes with the #-axis. Now suppose 8x and Sy 
to become indefinitely small. P and Q must approach 
indefinitely near to each other, the chord becomes a 

tangent, and -~~ = -&- 
PL ax 

We now have a geometric meaning for the first 

el v 
derivative: If y =f(x), -j- is the tangent of the angle 

which the tangent to the curve makes with the x-axis. 

The direction of the tangent determines the direc- 
tion of the curve at the point of tangency. The value 

of -j- at any particular point on the curve gives us, 

therefore, the slope or gradient of the curve at that 
point. 



THE GRAPH. 45 

If a = tan -1 -^, we also have, when P and Q are 

TO 7 

indefinitely near to each other, -7^ == -^ = sin a, and 

PL dx P $ ds 

~57] = 1T = cosa > ds being the elementary arc PQ. 

39. The student will at once perceive that the first 
derivative must be of great use in searching for special 
features of any graph. For one important applica- 
tion, let us see what it can tell us about the graph of 
ax + by + c = 0. Differentiating this expression, 

a + bf = 0, 

ax 



and therefore 



dy _ a 
dx b 



We have here a constant value for the tangent of the 
angle which the graph of ax + by + c = makes with 
the #-axis. Accordingly, the slope is constant and the 
graph can have no bends ; for a bend means change 
of slope. Therefore ax + by + c = must be a straight 
line and its own tangent. But ax + by + c = is the 
general equation of the first degree, and any property 
proved for it holds for any and every particular equa- 
tion of the first degree. For instance, the graph of 
y = x + 2, which seemed in Art. 36 to be a straight 
line, we now know to be a straight line. Further, 

dy 
from y = x + 2 w r e have -j- = 1. Since 1 is the gra- 
dient of this particular line, we know that it makes an 
angle of 45° with the horizontal axis. 



46 CALCULUS. 

Again, in the curve y = -, jr = 2' Here tana 

varies inversely as the square of the abscissa, and is all 
the time negative. It follows that at every point the 
tangent to the curve makes an obtuse angle with the 
#-axis. 

The angle a is always measured from the #-axis on 
the right-hand side of the origin, counter-clockwise 
around to the line which, with the x-axis, forms the 
angle. 

Exercises. 

40. 1. A point moves in a circle around the origin as a 
center, with a radius r. 

(1) The equation to the circle must be x 2 + ?/ 2 = r 2 ; for 
the abscissa and ordinate are all the while the sides of a 
right triangle. 

dy _ x 



(2) Show that 



ux -\/?~ — 



(3) Find the coordinates of the point or points where the 
circle has a slope of 1. 

2. Find the point of tangency when the tangent to y = - 

x 
makes equal angles with the axes of reference. 

Put -¥- = tan 135° and solve for x. 
dx 

3. Show that the curve y = goes through the 

1 + x 2 

origin. Find its slope at the origin. 

4. Construct the curve y 2 = 4 x. Find the point of tan- 
gency when the tangent to the curve makes an angle of 45° 
with the a>axis. 



THE GRAPH. 47 

5. Construct the curve y = sin x, making as much use as 

possible of -^ to determine the slope at various points. 

The x-axis must here be regarded the circumference of a 
circle whose radius is unity, straightened to a right line 
with the origin marked 0° . We easily obtain a num- 
ber of points on the curve by using the pairs of coordi- 
nates : 0°, 0; 45°, iV2; 90°, 1 ; 135°, i-V2; 180°, ; 225°, 
— |-V2, etc. Hence the curve passes through the origin, 
has a maximum ordinate at 90°, and crosses the x-axis again 
at 180°. In order to measure off the abscissas, the angles 
45°, 90°, 135°, etc., must be expressed in radians. We have 
the radian 57°.2958 ••• for the unit of distance. The abscissa 

indicated by 45°, for instance, is = — approxi- 

J ' ' 57.2958 191 Xi 

mately. The distance from the origin to the second point 

1 80 
of crossing is — ^ = 3.14159 •••, and the maximum ordi- 
nate therefore meets the o:-axis at a distance 1(3.14159 •••) 
from the origin. 

From 180° to 360° the values of the sines are a repetition 
of the values for the first semi-circumference, except that 
they are now all negative. Hence this portion of the curve 
is in every respect like the portion from 0° to 180° ; but it 
lies below the x-axis, and the direction of its convexity is 
reversed. 

Since sin(?i7r + x) = sin a?, n being even and positive, it is 
seen that the curve keeps its sinuous character, crossing the 
a>axis at regular intervals an unlimited number of times. 
On account of the repetition over and over again of the 
series of values of sin x, the function is called a periodic 
function. The curve itself is known as the sinusoid. 

6. Construct the curve y = cos a;. 

It is obvious in advance that this curve, which might be 
called the co-sinusoid, must be precisely like the sine curve 



48 CALCULUS. 

or sinusoid ; and that we shall have it in its proper position 
if we suppose the sine curve moved a distance of 90° to the 
left along the .T-axis. 

7. Find the first point to the right of the ?/-axis where 
y = sin x and y = cos x cross each other (see Art. 95). Show 
that the angle at which they cross is 180° — 2 tan -1 ^-V2. 

8. Construct y — m sin nx. 

9. Construct y = m cos nx. 

Assign numerical values to m and n\ then give a series 
of values to x, as in the first case. If a negative value is 
given to m, the effect is to rotate the curve on the a>axis so 
that portions which were above are now below, and vice versa. 

41. If ' = for some value of x, a = ; hence, to 

find whether the point describing a curve is anywhere 

dv 
moving parallel to the a>axis, we must put -~ equal to 

CtOu .. 

zero. Let .r x represent one root of the equation -^- = 0. 

If a value of x a little less than this root makes ~^- 

dx 

positive, and a value a little greater makes it negative, 
the tangent to the curve must make an acute angle with 
the #-axis, then become parallel to it, then make an 
obtuse angle with it ; and the curve must have a bend, 
being convex upward. The ordinate of the highest 
point, corresponding to x v is a maximum. So we 
define a maximum value of a function as a value greater 
than the value just before it and also than the one just 
after it. (PB, Fig. 3.) 

On the other hand, if -~ changes from — to + in 

passing through zero, the curve is concave upward, and 



THE GRAPH. 



49 



the lowest point is the end of a minimum ordinate ; that 
is, the value of the function is less than the value just 
before it and the one just after it. (P ; C, Fig. 3.) 

dv 
It is evident that if -j- is changing from + to — , 

— ; and if JL is changing from 
dx 



d [dy 
dx \dx. 
d (dy 
dx\dx 



is 



to +, 



is +. 



42. A third case arises : 



If -~ does not change siorn 
dx h h 



in passing through zero, there is neither a maximum 
nor a minimum ; but the point after reaching P or P f 
takes the path indi- 
cated by the dot- 
ted line. The point 

where -f- = is then 
dx 

called a point of in- 
flexion. In this case 

dx\dxj 

Every one is famil- 
iar with the point of 
inflexion as a feature 
in railroads, when 

the track is concave, say with respect to the fields on 
the right, and then changes so as to be concave to the 
fields on the left. Curves containing points of inflexion 
are very common in architectural forms. Such a curve 
is then known as an ogee. 

The same curve may of course have several maximum 
points and several minimum points, and also points of 




Fig. 3. 



50 CALCULUS. 

inflexion. Maximum and minimum points must evi- 
dently alternate. 

Exercises. 

43. 1. Consider the meaning of the statement ~ = cc. 

ax 

Examine the two cases : (a) when -— changes sign in pass- 

cioc 

(h/ 
ing through an infinite value ; (b) when -p does not change 

sign in passing through such a value. 

2. Examine the following curves for maxima and minima: 

(i) y = T^—>' (iii) y = ^ogx. 

JL - (- X 

(ii) -, + ^ = 1- (iv) y = 2px. 

or 0~ 

3. Draw the curve ?/ = e x , showing that it lies wholly 
above the as-axis, crosses the ?/-axis at an angle of 45°, and 
has no maximum or minimum points for any finite value 
of x. 

44. To illustrate the use of the principles established 
in Art. 41, suppose Ave know the slant height a of a 
right cone and wish to find the radius of its base when 
the volume is a maximum. Let y be the volume and x 
the base ; then 



TTX* 



— Va 2 — x 2 . 
J 3 

x and y, being mutually dependent variables, must ad- 
mit of graphical representation ; the abscissa of the 
point tracing the curve or graph is the varying radius, 
and the ordinate is the varying volume. Hence, if we 

put -~ equal to zero and solve the equation so formed, 



THE GRAPH. 51 

the value of x obtained will be the radius which gives 
the maximum volume. Differentiating 



y = Vr- z z 

U 3 



dx 


■K 

= 3 




IT 




"8 



- 2 # Va 2 — a; 2 + x 2 



-2x 



v2Va 2 — # 2 
2 # (fa 2 — x 2 ) — x s 



Va 2 — # 2 

putting this expression equal to zero, 

2x(a*-x*)-x s =0; 

hence x = V-| a. 

That is, the volume of the cone w T ill be greatest when 
the radius of the base is Vf a. 

In a case like this it is unnecessary to inquire whether 

-j- changes sign, and whether the change is from + to 

— or from — to + . For the volume of a cone of given 
slant height evidently varies from no volume when the 
radius is zero, through finite values to no volume again, 
when the radius is equal to the slant height ; that is, 
from a cone that is all height and no base to one that 
is all base and no height. Somewhere between these 
two extreme cases there must be a cone of ordinary 
shape whose volume is the greatest possible. It is well 
occasionally to supplement mathematics with common 
sense rather than to rely mechanically and invariably 
on some rule or formula. 



52 CALCULUS. 

Examples. 

1. ' Find the altitude of the right cylinder of greatest 
volume inscribed in a sphere whose radius is r. 

Alt. -I!. 

V3 

2. Given a point on the axis of the parabola y^—Xfx^ 
at the distance I from the vertex, rind the abscissa of 
the point of the curve nearest to it. x = I — 2 p. 

3. Find the maximum rectangle that can be inscribed 
in the ellipse whose axes are a and b. 

The sides are aV2 and 5V2. 

4. A talus resting on a horizontal plane has a slope 
of 30°; at the top of the talus is a series of strata 5 ft. 
thick ; the entire height of the ledge is 30 ft. How 
far must one stand from the foot of the talus to get 
the best view of the strata ? 

The angle at the observer's eye, formed by lines 
drawn to the bottom and to the top of the strata, must 
be a maximum. Let this angle be «; the angle sub- 
tended by the talus, /3, and the angle subtended by 
both talus and strata, 7. Also, let x be the horizontal 
distance from the observer to a point directly beneath 

25 30 

the strata. Then tan /3 = — ; tan 7 = — ; and 



x 



therefore tan a = 



30_25 

xx 5x 



750 x 2 + 750 ' 



d 5 (a? + 750) -10 a? 

and — tan a = — ^ — -+- ^— — 

dx (x 2 + 750) 2 



THE GRAPH. 53 

Equating this derivative to zero, x = 5V30, and 
finally the distance sought is 

5 V30 ^|— = 5 V3 ( VTO - 5). 

tan 30° v J 

5. The strength of a rectangular beam of given 
length, loaded and supported in any particular way, 
is proportional to the breadth of the section multiplied 
by the square of the depth. If the diameter a is given 
of a cylindric tree, what is the strongest beam which 
may be cut from it ? 

Let x be the beam's breadth ; then Va 2 — x 2 must be 
its depth. Hence, if y =x(a 2 — x 2 ), the strength is a 
maximum when y is a maximum. 

— = a 2 — 3 x 2 = 0, and therefore x = — — 
dx V3 

In the same way find the stiffest beam which may be 
cut from the tree by making the breadth multiplied by 
the cube of the depth a maximum. We now have 

y = x(a 2 — x 2 )* ; 

^ = ( a 2 _ 2.2)1 + z x Q a 2 _ ^)i(_ 2 x) = 0, and x = -. 
dx 2 

— Perry's Calculus for Engineers. 

6. The volume of a circular cylindric cistern being 
given (no cover), show that its surface is a minimum 
when the radius of the base is equal to the height of 
the cistern. 

Let x be the radius and y the height ; then the 
volume is irx 2 y, which equals a constant, say a. If S 
is the surface, 

S = 7rx 2 + 2 irxy = irx 2 -\ , since y = - — 5 - 

x irx 1 



54 CALCULUS. 

d S 

Finding — ■ and putting it equal to zero, we have 
ax 

TTX 1 *! 1 XT T , • 

-, and x — y. How do we know that tins 

IT 

makes the surface a minimum rather than a maximum ? 



x* — — = -, and x = y 

IT 



7. Determine the speed most economical in fuel to 
steam against a tide, supposing the resistance to vary 
as the nth power of the velocity through the water. 

Let a denote the velocity of the tide, x the velocity 
of the steamer through the water ; then x — a will he 
the velocity of the steamer relatively to the bank. 
The power required, and therefore the coal burnt per 
hour, will vary as the product of the resistance and the 
speed ; that is, as af +1 , and therefore the coal burnt per 

x' i+1 

mile will vary as This is to be a minimum, 

i t x — a 

hence we have 

d_( x n+1 \(n + l)x n (x -a)- x n+1 _ m 
dx \x — a J (x — a) 2 

, x _, 1 x — a 1 

and -=!+-, or = — 

a n an 

Thus if the resistance is taken to vary as the square 
of the velocity, the speed past the bank should be half 
the velocity of the current. 

— GreenhilPs Differential and Integral Calculus. 

8. Let A and B be two point-sources of heat. It is 
required to find the point M on the straight line AB, 
which is at the lowest temperature, the intensity of the 
radiation of heat varying inversely as the square of the 
distance from the source of heat. Let a be the distance 



THE GRAPH. 55 

between the points A and B, and x the distance from A 
of the point M on the straight line ; then 

AM = x, and BM= a — x. 

Let the intensities of heat at unit distance from the 
sources of heat be denoted bj^ a and /3 respectively. 
Then the total intensity of heat co at the point M 

will be 

a /3 

*> = ~n + 



x 2 ' (a — x) 2 
For a maximum or minimum, 



dco 
dx 



that is, 



and 



_2a 

£ 3 


+ - 


2/3 
2 — x) z 


(a — 


2:) 3 


/8 


a? 


I 


« 


a — 


X 


^8 



o, 



V # 



The distances i?ikf and Jl71T have, therefore, the same 
ratio as the cube roots of the corresponding heat 
intensities. 

Solving for x, 

_ aV« 

In this case it is necessary to see whether the value 
found corresponds to a maximum or a minimum. Dif- 
ferentiating the expression for — , we have 

dx 

d 2 co__2-3a 2-3/3 



dx 2 x^ (a — #) 4 



56 CALCULUS. 

which is positive for all values of .r, including the value 



a-fy" 



</a+</{3 



ay is therefore a minimum.* 

45. Suppose a line AB through the origin to revolve 
around counter-clockwise, making the variable angle 
with the .T-axis. Let AB pass through a point 
P(x, #), and let the distance of P from be denoted 
by r. r and are called polar coordinates; Ois the pole. 

Projecting OP onto the x and y axes respectively, we 
have x = rco$0 and y = rsin0. Through these rela- 
tions F (a?, y) = becomes .F(rcos0, rsin0)==O. For 
example, the equation oft + y 2 — 2ay = becomes in 
polar coordinates r 2 cos 2 # + / ,2 sin 2 — 2a(rsin#) = 0; 
that is, r = '1 a sin 0. This is readily seen to be a circle 
to which the x-axis is tangent, the point of tangency 
being the origin, or pole. 

Whenever any value of makes r negative, we meas- 
ure from the origin away from that end of the line AB 
which is tracing the arc that measures 0. If we im- 
agine an arrow to lie in the line AB and rotate with 
it, the barb may be regarded as tracing the arc that 
measures #, while the feather-end is negative. 

46. Let PP f be an arc 8s of a curve f(r, 0) = 0, PQ 
the arc of a circle whose radius is r ; and let the angle 
POP' = 80. In the limit 'PQP' is a right triangle, 
PQ = r d0, QP'^dr, and PP' = ds. Let <j> be the 

* Nernst and Schonflies' EinfuKrung in die mathematische Beliand- 
lung der Naturwissenschafien. 




THE GRAPH. 57 

angle made by the radius vector OP and the tangent 

to the curve ; then tan 6 = ~^~ = -=— . 

QP' dr 

Whenever the radius vector r is a maximum or mini- 
mum, the tangent at 
its extremity must 
be at right angles to 
it ; that is, 

rdO dr A 

dr rdd 

Points for which 

r is a maximum or 

minimum are called o A 

apsides. To find, / FlG 4 

therefore, whether 

I u r 
a given curve has an apsis, we must put = and 

solve this equation. 

For example, let us take the polar equation to the 

ellipse, the pole Tbeing at the right hand focus (see 

Art. 115). 

_ a (1 — £ 2 ) -j dr __ a (1 — 6 2 ) e sin 6 u 

1 + e cos ff an Id ~ (1 + ^cos^) 2 ; 

1 dr & sm 

then — 77T = q n\ and equating this to zero, 

r dO 1 + e cos ^ 6 

sin 6 = 0. Hence, the apsidal values of are 0° and 
180°. These results agree with what we observe in an 
examination of the given equation to the ellipse : r is a 
maximum, a(l + g), when 0=180°, and a minimum, 
a(l -O, when = 0°. 

The student who is unacquainted with the formal analytic 
geometry of the straight line and the conic section is advised to 
read Chapter IV before beginning the next chapter. 



CHAPTER III. 

APPLICATIONS. 

47. In the mathematical sciences one of the most 
common of fundamental variables is time ; and when the 
function of time is the space passed over by a body, the 

first and second derivatives — and — ( — - ) are of great 

, tit dt\dt h 

importance. N 

Suppose a body moves over equal spaces in equal 

times. The space divided by the time gives the speed 

or velocity of the body. That is, if s is the space passed 

over in the time t, - is the velocity of the body. 

" While the camels were being loaded, T measured my first base- 
line of 400 metres. Boghra (my riding camel) walked it in five 
and one-half minutes. This was a daily recurring task, for the 
contours of the ground varied a good deal, and the depth of the 
sand made a very appreciable difference in the time the camels 
took to do the same distance." 

— Sven Hedin's Through Asia, Vol. I, p. 482. 

In this illustrative case, - = — — = the speed of the 

t 51 

camel expressed in metres per minute. Assuming that 

o 

- was a constant during each day, the distance travelled 

u 

on any given day by Hedin's caravan was known by 
multiplying the speed by the time spent in travel. 

58 



VELOCITY. 59 

48. If the motion is variable so that the body does not 
move over equal spaces in equal times, we may obtain 
an expression for velocity by taking the time so short 
that during that time the motion must be uniform. So 
if dt be an indefinitely short time and ds the indefinitely 

small space passed over in that time, — is the velocity 

and is measured by the space that would have been 
passed over in a unit of time if the body had kept on 
moving for a whole unit with the velocity which it had 
at the instant considered. For instance, if we say that 
a train is running at the rate of 30 miles an hour, we 
mean that if it were to run for a whole hour with the 
same speed which it has at this instant it would pass 
over a distance of 30 miles. As a matter of fact it may 
stop in a few minutes ; that has nothing to do with its 
speed at this instant. But 30 miles per hour is the same 
as 1 mile in 2 minutes, or 4.4 feet in .1 of a second, and 
so on. Evidently the rate remains the same so long as 
the ratio of the space to the time is the same, however 
small the space and the time may be individually. 

Hence, in this case, — = 30 miles per hour. 
dt L 

If we know the whole space passed over by a body 
and know also the time taken, the space divided by the 
time is the average velocity: it must not be confused 
with the velocity proper, which may have varied during 
the time. For example, the first mail cartridge sent by 
compressed air from the Boston post office to the North 
Union Station (Dec. 17, 1897) required 1 minute and 
2 seconds to pass from one place to the other, a distance 
of 4500 feet. The average velocity was 72.58+ feet 
per second. 



60 CALCULUS. 

49. If a body is moving in a northeasterly direction, 
it plainly has a motion eastward and a motion northward. 
For instance, if it is moving due northeast with a ve- 
locity of 20 miles per hour, it is getting eastward at 
the rate of 20 cos 45° miles per hour, and northward 
at the same rate. If it is moving east 30° north at the 
rate of 20 miles per hour, it is moving east at the rate 
of 20 cos 30° miles per hour, and north at the rate of 
20 cos 60° miles per hour. 

In general, if a body is moving with a velocity v along 
a line which makes with the .r-axis an angle of a degrees, 
its component velocity parallel to the #-axis is v cos a, 
and its component velocity parallel to the y-axis is 

v sin a. We have already seen (Art. 38) that — = cos a 
7 7 ds 

(1 1J Ci S 

and — ^ = sin a. Hence, if -— is the velocity of a body at 
ds at 

, . CIS (XX CtX -\ CtX ■ ,i c 

any instant, v cos a = — . — - = — ; and — - is theretore 
dt ds dt at 

the component velocity parallel to the #-axis. Similarly, 

v sin a = — . — ^ = -^ = the component velocity parallel 
dt ds dt r J L 

to the y-axis. 

Evidently a velocity parallel to any line furnishes 

a component velocity parallel to any other line if it 

be multiplied by the cosine of the angle between the 

lines. 

50. Suppose a particle is moving in a plane curve and 
we wish to know its component velocities at any instant 
(1) along the radius vector, and (2) perpendicular to 
the radius vector. 

We have x = r cos 6 and y = r sin 0, in which x, y, r, 



VELOCITY. 



61 



and 6 depend upon the time t. Differentiating with t 
as the fundamental variable, 



dx _ dr 
di~~dt 



cos i 



sin 6 



dd 

dt 



dy dr . a , add 

-2- = — sin u + r cos u — 

dt dt dt 

dx 



O) 



dx 
According to the preceding article, — is the velocity 



dx 



parallel to the #-axis and ^ cos 6 is the component 
which it furnishes along the radius vector. Similarly, 
-2 sin 6 is the component which -2 furnishes along the 
radius vector. The sum of these components is the 




Fig. 5. 



whole velocity along the radius vector. From equa- 
tions (a) and (b) we have 



dt dt at 



O) 



^COS0- 


dx . n dO 
sm v = r — 


dt 


dt dt 



62 CALCULUS. 

Again, resolving along a line perpendicular to the 
radius vector and combining the parts, 

oo 

The reason for the minus sign in the first member 
of equation (Y?) should be noticed. The velocities 

-f- cos 6 and — sin 6 are oppositely directed (see 
dt dt 

Fig. 5); hence, when combined, their difference must 
be expressed. 

51. -t\-t\ the rate of change of a variable velocity, 

is called acceleration. 

— ( — ] and — ( — ) are the accelerations parallel to 
dt\dt) dt\dtj l 

the #-axis and ?/-axis respectively ; and we can now find 
the component accelerations (1) along the radius vector, 
and (2) perpendicular to the radius vector. 

Differentiating equations (a) and (J) of the preced- 
ing article, 



d 2 x 

It 2 '' 

d 2 y 
It? 



— --r — - cos0 — 2 — — +r— — sin (9, (e) 
_dt 2 \dt) J \ dt dt dt 2 ) 

r ( sm 0- 2- — + r — cos0. (/) 



_dt 2 \dtj 



dt dt dt 2 ) 



Multiplying equation (/) by sin #, and equation (e) 
by cos 6 and adding, we have 

for the acceleration along the radius vector. 



ANGULAR VELOCITY. 63 

Again, multiplying (/) by cos (9, and (e) by sin 
and subtracting the latter product from the former, we 

d 2 y « d 2 x . n dr d6 , d 2 , 7 , 

—f cos - — ■ sin = 2 — — + r — - (h) 

dt l dt 2 dt dt dt 2 

for the acceleration perpendicular to the radius vector. 
It is to be noticed that the second member of equa- 
tion (li) may be written - • — ( r 2 — )• 
J r dt\ dt) 

52. Angular velocity is defined as the ratio of the 
angle differential, d0, to the time differential, dt. This 

d0 
ratio, — , may be a constant or a variable. For ex- 
cic 

ample, the earth rotates on her axis with constant 

angular velocity, and — = — — ; but she moves in her 

s J dt 2-i h 

orbit around the sun with a variable angular velocity. 
(See Art. 88.) 

53. As an important application of the results given 
in equations (</) and (K) above, suppose a particle is 
moving in a circle with constant angular velocity. 

Then, since r is a constant, — =0 and — ( — ) = 0. 

dt dt\dtj 

Therefore the acceleration along the radius vector re- 
duces to — ri— ] . Also, in equation (A), since — = 0, 

the term 2 is zero; and since — is a constant, 

— [ — ) = 0, and the term r — - is zero; and therefore 
dAdtJ dt 2 

the acceleration perpendicular to the radius vector is 
zero. This conclusion is what we might have expected 



64 



CALCULUS. 



from the premise that the particle moves in a circle 
with constant angular velocity. 

f ao\* 



54. The above expression, — r ( — ) , for the accelera- 
tion along the radius vector when a particle is moving in 
a circle with constant angular velocity, may be written 



r \dt 



or 



1 (rd0\* . 
r\dt) ' 



but since rd0 is the length of the arc corresponding to 
the angle dd, is the linear velocity v of the particle. 

Lit o 

Hence we have as a simple form for the accelera- 

r 

tion along the radius vector when the particle or body 
moves in a circle with constant angular velocity. 

55. Suppose a point Q moves with constant angular 

d0 
velocity — - or co in a circle AQA f whose radius is r. 

Take the center as or- 
igin, and let QP be 
the perpendicular from 
Q to the j/-axis OA. 

As the angle AOQ 
increases, the line QP 
increases from to r 
and then decreases. 
The changes in the 

ratios % % etc -' 

due to changes in the 
angle AOQ, have already been discussed (Art. 20). 
We shall now consider the motion of the point P as 




Fig. 6. 



SIMPLE HARMONIC MOTION. 65 

Q describes the circle. OP is the ordinate of P at 
any instant, and if Q has taken the time t to move 
from A to Q, the angle AOQ = cot ; hence 2/ = r cos o>£. 
If $ starts at some point #', and £ is the time required 
to move from Q 1 to A, the angle #' OA = cot ; hence, 
counting the time from the start at Q', the angle Q' OQ 
= cot and the angle A OQ = cot — cot ; and therefore 

y = r cos (jot — o)£ ) = r cos (o>£ + e) 

if we write e for the constant, — cot . 

56. In regard to the motion of P, we notice at once 
that it must cross the circle on the diameter AA f and 
return to A in the time that Q is describing the circum- 
ference ; so its motion is vibratory. It starts with zero 
velocity, and must be going with its greatest velocity 
when at the center; for its direction of motion is then 
parallel to that of Q. 

To get a more precise knowledge of the motion of P, 
let us take e = 0, so that y = r cos cot. By doing this 
the equation gains in simplicity and the motion remains 
the same, but the time is counted from the instant when 
Q is at A instead of Q f . 

We now have y = r cos cot, (a) 

-Jl = — rco sin cot. (b~) 

dt V J 

-JL = — rco 2 cos cot. (c) 

dt\ 

Equation (6) shows that the velocity of the point is 
greatest when cot = 90° ; that is, when P is at the cen- 



66 CALCULUS. 

ter. Equation (V) shows that the acceleration is great- 
est when cot = 0° and 180° ; that is, at the start and at 
A! . Also, the acceleration is least when cot = 90°. 

57. The variation in the ordinate OP may be best 
appreciated by noticing the identity of the equation 
y = r cos cot with the equation y = m eosnx given in 
exercise 9, Art. 40. Equation (a) accordingly repre- 
sents a cosine curve. Further, if the velocity equation 
(6) be graphically shown, its curve must be the sinus- 
oid. And finally, the acceleration equation (<?) is 
another cosine curve, differing from the first, how- 
ever, on account of the coefficient — r&> 2 , which has 
replaced the coefficient r in equation (a). 

It is w^ell worth the student's while to construct care- 
fully the graphs for the three equations (a), (/>), (V), 
using the same unit of length for all three. The usual 
#-axis now becomes a time axis in each case, sii\ce the 
abscissas are times. The y-axis for (a) is a displace- 
ment axis ; for (J) it is a velocity axis ; and for (c) an 
acceleration axis. 

58. We are now familiar with the geometrical mean- 
ing of —^ when y =/(#). If y =/(£), — ^ is analogous 

7 (XX Ct'L 

to -^, and must have the same geometrical meaning. 
dx 

That is, viewed geometrically rather than kinematically, 
-^ is the tangent of the angle which the tangent to the 

Ctv 

curve y = f (t) makes with the £-axis. Accordingly, 
equation (6), Art. 56, might be called the curve of the 
tangent to (a) ; for any ordinate (with the abscissa t f ) 



SIMPLE HARMONIC MOTION. 67 

in the graph of (5) represents the magnitude of the 
slope of (a) at the point whose abscissa is t'. Evidently 
the curve of (c) is related to (6) just as (&) is to (a). 

59. The point P, vibrating back and forth across the 
circle (Fig. 6), is said to have simple harmonic motion. 
It is such motion as this that Jupiter's satellites seem 
to have as we look at his orbit " edge on." 

The range OA or OA! on one side or the other of 
the middle point is called the amplitude ; and the ordi- 
nate OP is the displacement. The period of a simple 
harmonic motion is the time which elapses from any 
instant until the point moves again in the same direc- 
tion through the same position ; that is, the time 
required by P to move from P ! to A! , thence back to J., 
and finally to the initial position P f , is the period. The 

phase is the fraction - — of the period of vibration. 

2 7T 

The epoch is the angle e. 

"This expression y = r cos (wt + e) is to be found, perhaps more 
frequently than any other, in all branches of mathematical physics. 
It is in terms, or series of terms, of this form that every periodic 
phenomenon can be described mathematically. From the expres- 
sions for the longitude and radius vector of a planet or satellite to 
those of the most complex undulations, whether in water, in air, 
or in the luminiferous medium, all are alike dependent upon it." 

— Tait's Dynamics. 

Example. Find an expression for the up and down 
motion of the connecting-rod of a locomotive. 

60. The downward fall of an unsupported bocly is 
due to the accelerating force exerted by the earth and 
known as gravity. At small distances above the earth's 



68 CALCULUS. 

surface this force is practically constant ; the accel- 
eration caused by it is denoted by g. When g is de- 
termined at different places on the earth, it is found 
to vary within narrow limits. This variation is due to 
several causes, the chief one being the rotation of the 
earth on its axis, g has its least value at the equator 
and its greatest value at the poles. At Washington, 
D.C., g is 980.098 dynes ; * that is, the observed accel- 
eration due to gravity is, at that point on the earth's 
surface, 32.155 feet per second. 

Taking the origin at the point from which a body 
falls, with the positive end of the ^-axis downward, we 

now have 

a fdy\_ 

Jt\dtJ 9 ' 

therefore, after integrating, 

cl l = v = gt+ C. (Art. 15.) 

at 

If the body falls from rest, v = when t = ; there- 
fore (7=0, and the equation becomes 

!-* w 

Multiplying by dt and integrating again, 

y = ±Cjt*+C l . 

Since y = when t = 0, O f = ; 
therefore y = \ gt 2 . (5) 

* U. S. Coast and Geodetic Survey. 



FALLING BODIES. 69 

Combining equations (a) and (b) so as to eliminate t, 

Equation (e) enables us to find the velocity with 
which a body is moving when it has fallen through a 
given space. For example, the monument at Washing- 
ton is 555 feet high ; if a ball is dropped from the top, 
what is its velocity upon reaching the ground ? We 
may take g = 32, a value sufficiently accurate in this 
example and similar ones. Then v = 8V555 = 188 feet 
per second, approximately. 

61. If the body is projected directly upward, 

dt\dt) *' 

because the acceleration is now a retardation tending 
to diminish y. Integrating as before, 

If the body is projected with the velocity V, ■-$- = V 
when t == ; therefore C = FJ and the equation becomes 

Multiplying by dt and integrating again, 
and since y = when t = 0, (7' = ; and we have 



70 CALCULUS. 

Combining equations (J) and (e) so as to eliminate t, 

62. By means of the equations of the two preceding 
articles we can readily show that if a body is projected 
vertically upward, it takes the same time to come down 
that it does to go up ; also, upon reaching the point 
from which it was projected, it has the same velocity as 
that with which it was projected. 



From (d), 


when 


dt g 


time up ; 


from CO, 


when 


dt ' V 2g 


space up ; 


from (5), 


when 


-0 9 


time down ; 


from (e), 


when 


V 2 

2 9 





In the derivation of formulas (a) to (/), no account 
has been taken of the resistance offered by the air to 
the fall or rise of a body. The formulas are strictly 
true only on the supposition that the acceleration is 
constant, and that the motion takes place in a vacuum. 

63. We may now consider the case when the height 
is so great that the acceleration cannot be regarded as 
constant. What is the velocity of the body on reach- 
ing the earth ? 

A homogeneous sphere, or a sphere composed of con- 
centric layers with the density varying only from one 
layer to another, attracts an external body with an 



FALLING BODIES. 71 

intensity varying inversely as the square of the dis- 
tance of the bocty from the center of the sphere. Let 
g be the acceleration due to the earth when the body is 
at the earth's surface, and/ the acceleration at the dis- 
tance y from the center. (Notice that the center thus 
becomes the origin.) 

Then, if R is the earth's radius, 

gB? 





9' 


R2 +1 f 
= — k ; that is. 

y 1 


./ = 


and therefore 


we 


have 








d ( d y\ _ 

dt\dt) 


V 2, 



The minus sign is taken because y is diminishing as 

the body falls ; that is, dy is negative, and since -^ is 

increasing numerically, -q-\-j~) must also be negative. 

If we multiply by dt, as in the previous articles, and 
attempt to integrate, we have 

dt~J y* dt ' 

an indicated operation which cannot be performed un- 
less we know what function y is of t ; and this we do 
not know in advance. But multiplying by dy instead 
of dt, 

d fdy\ __ dy fdy\ __ gR 2 



d n\tt) = iti d Kdtr-^ cly - 

The first member is immediately integrated by ob- 
serving that it is of the form xdx, and that 



/■ 



JU CI Jb c) 



72- CALCULUS. 

We have, therefore, 



gjr 



If the body falls from the height h above the earth's 

(Jb II (1 RP* 

surface so that y = R + h when -^- =0, C = — -~ — r > 

at R + h 

and the equation becomes 

yw= qm (\ L_ 

2\dtJ y \y R + h 

The same result is reached by writing a definite 
integral (Art. 16) whose limits are R + h and y. We 
then have 



1W = p _/z^ : 



X 



2\efc/ «^+a */ 2 



= -<^ 



dy 
1" 



= -^ 2 - = + ■ 



^ 2 



# -B + A. 



\y R + h 



Suppose that 



dy_ 

dt 



v l when y = R\ that is, v 1 is the 



velocity which the body has when it reaches the earth's 
surface. Then 



iw=#-„- 



R R + h 



= gR 



R + h 



■ gh 



R 



R + h 



FALLING BODIES. 73 



and therefore 



^V^-i+I- 



r m 

If h<R, the series within the parenthesis is converg- 
ing ; and if h is very small in comparison with 72, we 
may drop all terms of the series after the first term ; 
we then have v f = V2^A, which is identical with for- 
mula (Y), Art. 60. 

If A>_R, the series is diverging; the formula con- 
taining it cannot therefore be used, and we return to 
one of the other expressions. For example, suppose a 
body falls from an indefinitely great distance ; what 
will be its velocity on reaching the surface of the 
earth, all forces besides the earth's attraction being 
disregarded ? 

We have £<V) 2 = ^ 2 (| - ~ A 



or v r = ^J2gR, 

when h is indefinitely great. 

If R = 3960 x 5280 feet and g = 32.155, 



v , = V2(32.155)(3960)(5280) ^ secQ 
5280 L 

= 7 miles per second, approximately. 

By " great heights" we may mean such various 
heights as those attained by the kites flown at the 
Blue Hill Observatory (8000 ft,), or by Andrews bal- 
loon; or the height of a meteorite when it first becomes 
visible. In the practical consideration of the velocities 



74 CALCULUS. 

of bodies falling from such heights, the resistance of 
the air must be taken into account. For a discussion 
of the vertical motion of a body in a resisting medium, 
see Greenhill's Calculus, Art. 7(5. 

64. Let a body free to move be subjected to an 
attractive force that varies directly as the distance of 
the body from the point where the force is located. 
If we take this point as origin, with a line passing 
through the body for the ^-axis, we have 

d fdx\ 

The coefficient /jl is seen to be the value of the accel- 
eration when x=l ; that is, when the body is at a unit's 
distance from the origin. The minus sign is used for 
the same reason that was given in Art. 63. 

Multiplying by dx and integrating, 

lA7.r\ 2 __^ 2 , c 
2\dt) ~ 2 X +C - 

If — = when x — a, C = *—- ; 
dt '2 

therefore -(— j =-(a? — x 2 ). 

Writing this equation so that dt shall stand by itself, 

dt=- 1 dx 



V//, Va 2 — x 2 

After extracting the square root only the negative 
sign is retained ; because dt is positive, and dx is nega- 
tive when the body is moving toward the origin. 



BECTILINEAR MOTION. 75 

Integrating again, 

V/x a 

If t = when x = a, O r = ; 

1 # 

therefore £ = — - cos -1 - ; 

V> « 

that is, # = a cos V/x£. 

Comparing this result with equation (a), Art. 56, we 
conclude that a body subjected to an attractive force 
varying directly as the distance will move with simple 
harmonic motion. 

65. Suppose the body is driven away from the origin 
by a force varying directly as the distance of the body. 

and proceeding as before, 

I — - J = /jlx 2 + C = fji (x 2 — a 2 ) ; (a) 

that is, Vfidt 



-Vx 2 — a 2 
Integrating again, we have 

tVJi+ C r = log (x + Vx 2 - a 2 ) . 

Notice that the constant of integration m&y be written 
in either member of the equation as suits our conven- 
ience. Heretofore it has been written in the right-hand 
member. 



76 CALCULUS. 

Now suppose that x = a when t = ; 

then . C f = log a, 

and tVfi + log a = log (x + V^ 2 — a 2 ), 

, /- i (x + Va 2 — a 2 \ 

^V/x = log — ]. 

\ a 

From this expression we have 



„<Lt 



x + V.t' 2 — a 2, = ae 
Further, since (x + Vx 2 — a 2 )(# — V# 2 — a 2 ) = a 2 , 



a; — V.r 2 — a 2 = — — = ae y//x '. 

Adding the expressions for 

x + VV 2 — a 2, and .r — V x l — a\ 
2x = tuft* + ae'^; 

that is, x = a - (e^ f + e"^). (&) 

If we now differentiate this expression, we shall have 
the velocity a function of the time instead of a function 
of the distance as in equation (a) ; for 

dt 2 K ^ >* J 

2 

Equations (5) and (c) show that as t increases, the 
body is driven farther and farther from the origin with 
ever increasing velocity. These equations involve the 



RECTILINEAR MOTION. 77 

supposition that the initial velocity is zero. Let us now 
suppose that the initial velocity is — aVft. Resuming 
the equation , 7 xo 

since [_p\== — aVa when x = a. (7=0: 

\dt) 

and the equation becomes 

fdx^ 2 



dx /— , 

or — = — V udt, 

x 

the minus sign being used because the motion is toward 
the origin. 

We now have log x = — Vfit + G\ 
and since x = a when t = 0, C f = log a ; 

therefore — V/jit = log --> 

and x = ae'^* 1 *. 

This equation shows that with the initial velocity 
— a V/x the body constantly approaches the origin, but 
never reaches it. 

66. Suppose that a body instead of being projected 
vertically, is projected in a direction making the angle a 
with the horizontal plane, V being the velocity of pro- 
jection. The body thus has a vertical velocity and a 
horizontal velocity. The horizontal velocity is evi- 
dently unaccelerated, whilst the vertical velocity is 
being retarded by gravity. That is, taking the hori- 



78 CALCULUS. 

zontal side of the angle a for the #-axis, and taking the 
y-axis vertical and positive upward with the point 
from which the body is projected as origin, 

d fdx\ _ n . d fdy\ _ 
dt\dtJ~ ; It\di)~ 9 ' 

These two statements are the " equations of motion " 
of the body. Examples of such equations have already 
occurred in preceding articles. Integrating the first 

(1 v 

one, — = J^cos a, the constant horizontal velocity. In- 
tegrating again, 

x = tv cos a, {a) 

the constant of integration being zero if t = when 
x= Q. 

Integrating the second equation of motion, 

When t = 0, the time of projection, -^ is the vertical 

1 J dt 

component of the velocity for the same instant. This 
initial vertical velocity being F~sin a, we have 

— ^ = — at + F"sin «, 
dt J 

and integrating again, 

y = — \gt 2 + t V sin a. (J) 

Equations (a) and (b) give the coordinates of the body 
at any time t. Eliminating £, we have 

y = x tan a - f x\ (» 

2 F 2 cos 2 a 

the equation to the path of the body. 



PARABOLIC MOTION. 79 

67. If we transform equation (e) by passing to a 
new pair of axes parallel to the first with 

V 2 sin a cos a V 2 sin 2 a 
9 2 9 

for the coordinates of the new origin, we have (Art. 100), 

Y^sin 2 a f , F 2 sin a cos a 

y H - = tan a[ x -\ 

2 9 V g 

g_ f F" 2 sinacos^ 2 



2 V 2 cos 2 a\ g 

After reduction this becomes 

« 2 F 2 cos 2 a 
x 2 = y, 

9 

which is seen to be a parabola convex upward with its 
vertex at the origin of coordinates. (Art. 129.) 

This curve is approximately shown in a stream of 
water issuing from a hose. It may also be traced by 
watching a tennis-ball or base-ball as the ball moves 
through the air. 

68. To find the horizontal range, we put y = in 
equation (<?); then x = — This value is great- 
est when sin 2 a is greatest; that is, when «, the angle 
of projection, is 45°. 

It may be noticed that persons skilled in throwing have learned 
from experience that in order to throw as far as possible the ball 
or stone must be thrown in a direction about half way between 
horizontal and " straight up." 



80 CALCULUS. 

69. To find the range on an inclined plane let the 
straight line y = x tan /3 express the slope of the plane. 
We have then to find where the line y = x tan /3 cuts 
the parabola 

y = x tan a - / x\ 

I V* cosset 

Eliminating y, we obtain 

_ 2 V 2 cos a sin (a — /3) 
g cosp 

the abscissa of the point of intersection. The distance 
from the point of projection to this point of intersec- 
tion is therefore 

Q t • •, ^ 2V 2 cos a sin (a — ff) 

#sec/3, which equals — — • 

g cos 2 ft 

To find the particular value of a that will make this 
distance a maximum, we must view this expression as 
a function of a and equate the first derivative to zero ; 

that is, if R is the range, - — ■ = is the condition for a 

da 

maximum (or minimum). (Art. 41.) 
We have then 

dR d n 

— = — x sec p 
da da 

— 2 V 2 sin a sin (a — /3) + 2V 2 cos a cos (a— /3) 
g cos 2 ft 
Equating this to zero and reducing, 

cos a cos (« — /3) — sin a sin (a — /3) = ; 
that is, cos [« + (« — /3)] = 0, 

and hence 2«-/3= 90°, 

90°+/3 



a = 



£-1-1(90°-/?). 



MOTION IX A VERTICAL CURVE. 81 

Therefore the direction of projection which secures 
the greatest range on a given inclined plane bisects the 
angle between the vertical and the inclined plane. 

The student should of course raise the question: 
How do we know that the above equation of condition 
gives a value of a that secures a maximum range instead 
of a minimum ? 

70. Suppose that a piece of smooth Avire or small-bore 
tubing, smooth on the inside, is bent into the shape of 
some plane curve and hung up vertically. Further, 
suppose that a bead* is strung on the wire, or a small 
ball dropped into the tube. The body, say the ball, 
will slide downward under the action of gravity, but 
it will be obliged to follow a certain path. What will 
its velocity be at any point P ? 

Draw the usual axes in the vertical plane in which 
the curve lies. Let A be the position of the body when 
£=0; P its position (x, y) at any time t\ and let 
arc^LP = s. If a is the angle which the tangent at the 
point P makes with the #-axis, g sin a is the acceleration 

along the curve at P. But sin a = — f- ; hence 

as 

d fds\ _ _ cly 

Jt\dtJ~ 9 Ts 
Multiplying by 2 ds and integrating, we obtain 

r ds^ 2 



dtj 

If we call the ordinate of the point A y and the 
velocity at A v , we have 

and therefore v 2 — v* = 2 g (y — g) . 



82 CALCULUS. 

Now let the ordinate t/ be produced upward to a 
point B, making AB = A, the height from which the 
body falling freely would have to fall in order to ac- 
quire the velocity v . Draw BN a line parallel to the 
#-axis. Let C be the point where the ordinate y pro- 
duced meets the line BN. v 2 =2</h (Art. 60 (<?)). 
Substituting this value of v 2 in the equation above, 

v*= 2gh -<Lg{y- y )= 2g(h + y -y-) = 2g ■ PC. 

Hence the velocity at any point P is the same as the 
velocity that would have been acquired had the body 
fallen directly from the line BN to P. 

71. Let us now limit the case to motion in a vertical 
circle. Instead of having the ball slide in a circular 

tube we can just as well 
secure circular motion by 
attaching the ball to the 
end of a string whose other 
end is fastened at the cen- 
ter of the circle. We now 
have a pendulum. Let C 
be the center of the circle ; 
Fig. 7. its lowest point ; OX the 

#-axis, and OY the y-axis. 
Let A be the starting point of the ball ; then at A t = 
and v Q = 0. Let P be its position and v its velocity at 
any time t. Also, let 6= angle P CO and a = angle A 00 ; 
s = arc AP and I = PC, the length of the string. 
By the preceding article, 

fd*\ 2 = v * = 2g . PJST= 2#Z(cos - cos <*); 




SIMPLE PENDULUM. 83 



but (*Y-Pf^' 



\dtj \dt. 

hence [ — - ) = -3- (cos — cos a) 

4gf . 2 a -off 
= —^ sm z snr - 

I \ 2 2 

and therefore — = - 2\k Jsin 2 - - sin 2 £ 
dt * l* 2 2 

Notice that after extracting the square root only the 
minus sign is used, because dt is positive and d0 is 
negative. 

We now have 



Vf^=- 



dd 



\|sin 2 ~ — sin : 



that is, 



>f— r 



2 



i\/sin 2 



_:_»0 



The expression here presented for integration looks 
quite simple, but it cannot be expressed in finite terms by 
means of the ordinary algebraic or trigonometric func- 
tions. If, however, we expand sin 2 - by Maelaurin's 

2 

theorem (Art. 34, ex. 14), and then take a so small 
that we may neglect powers of a (and 0) beyond the 
second, we shall have 

4j"sin 2 |-sin 2 |) = « 2 -^ 2 . 



84 CALCULUS. 

The above integral then becomes 

and this is integrated by formula ll r Chap. V, so that 
we obtain 



C 



Vf- 



a 





= cos" 1 cos -1 - = cos L - 

a a a a 



Solving for 0, = a cos y k 

When (9=0, -^2* = cos^O = -; 

hence f , the time from J. to (9, is —\— 

2 V 

If T 7 be the time of an oscillation from J. to A 1 (on 
the other side of 0), 

This result is true only when a is small, as above 
shown. It is independent of «, and therefore the time 
of an oscillation is the same for all small arcs in the 
same circle. That is, if a and a! are two small but 
unequal arcs, the times of oscillation for the same 
pendulum are equal. 

72. It will be noticed that the equation 
6 = a cos\f- t 



AREAS. 



85 



is of the form of the equation expressing simple har- 
monic motion ; therefore the pendulum-bob. has simple 
harmonic motion in a circle which lies in a plane pass- 
ing through OX perpendicular to OF. The radius of 
this circle is «, and the displacement at any time t is 6. 
If a is given in degrees, it must be divided by the 
radian (57°. 295779 •••). For instance, if I = 50 inches 
and a = 1°, the radius of the circle across which the 

harmonic motion takes place is — - — i- = — inches, 

. , , L 57.°+ 57 

approximately. 



73. Areas. Let PS be a portion of the curve 
y =f(x) ; and let it be required to find the area 
bounded by this arc, the ordinates PM and SN, and the 



#-axis. 




Fig. 8. 



Let OM=a, ON=b, OT=z, and OV=z + 8p; 
then QT=y, and RV=y + 8y. If the area OLQT, 
any varying portion of the area OLSN, equals ^4, area 
OLR V=A + 8 A, and 8A = TQR V. Now, if the short 



86 CALCULUS. 

arc QR were a straight line, the area TQRS would be 
a trapezoid, and we should have 

8A = Sx±(QT + RV)= Sx(y + ±8i,) ; 

and — = y + \ Si/, 
ox 

In the limit QR becomes a straight line, and 
dA 

that is, dA = ydx = f(x)dx ; 

and this is a representative strip taken anywhere in the 
area OLSN. 

Suppose \f(x)dx = <f>(x) + 0; 

then A = 4>(x)+C. 

Since we are measuring areas from the y-axis, 
when x = 0, A = ; 

when x = a, JL = area OLPM; 

when # = 6, -4 = area OLSN; 

therefore area OLSN= <£(J) + (7, 

area OLPM=<t>(a) + O. 

Subtracting this last expression from the one pre- 
ceding it, 

area OLSN- area OLPM= mmMPSN 



f(x)dx. 

a 



ABE AS. 87 

We have, then, for the area between the ordinates, 
whose distances from the ?/-axis are a and b respec- 
tively, the definite integral 



r 



f(x)dx. 



74. For example, suppose we wish to find the area 
bounded by the parabola y 2 = \px, the a>axis, and any 
ordinate y r (the accompanying abscissa being x' ). 



A= Cydx = C x 2p i x*dx: 



4:p 2 X 2 

3 



4 /V)2/v^2 



We notice that the rectangle x'y f = 2p 2 x' 2 ' ; hence 
the area in question equals two-thirds the circumscribed 
rectangle. 

Examples. 

1. Find the area of the upper right-hand quarter of 

the ellipse •— + &-■ = 1. 



In this case 



o ydv=J ~^Ja 2 -x l dx 

h \ x r* 2 , a2 • -l ^~ 

_ 7raS 

The area of the entire ellipse is therefore it ah. ira 2 , 
the area of the circle x 2 + y 2 = a 2 , may now be viewed 
as a special case of the ellipse in which b = a. 



88 CALCULUS. 

2. Find the area between the isotherm pv = c, the 
v-axis, and the two ordinates whose distances from the 

#-axis are a and b respectively. Aits, c log — 

a 

3. Find the area bounded by the #-axis and the 
curve y = sin x, from x = 0° to x = 180°. 

Ans. 2.. 

75. Mean values. The mean or average value of n 
quantities is the nt\\ part of their sum. If the quanti- 
ties to be averaged are successive values of a function 
of some variable, their magnitudes depend not only on 
the nature of the function, but also on the law of varia- 
tion of the fundamental. Thus, suppose we have the 
isotherm pv = c and wish to know the average pressure 
between the volumes i\ and v 2 . It is necessary to make 
some assumption in regard to the variation of v. (1) If 
its increments are supposed equal, we understand by the 
"mean value" of the pressure the average of the press- 
ures corresponding to the arithmetic series : v, v + civ, 
v + 2 civ, etc. (2) If the volume is assumed to depend 
on some other variable in such a manner that the 
abscissa increments are not equal, the mean value will 
now be the average of a new series of pressure ordinates 
corresponding to the new series of values of v arising 
under the second assumption. Evidently the two means 
will, in general, be unequal ; but one is just as properly 
the average as the other. An important illustration is 
afforded if we ask : what is the mean distance of a 
planet from the sun ? If a planet moved in its elliptic 
orbit in such a way that the radius vector described 

equal angles in equal times, that is, if — , its angular 

at 



MEAN VALUES. 89 

velocity, were a constant, the mean length of its radius 
vector could be shown to be aVl — e 2 , a being the semi- 
major axis, and e the eccentricity of its orbit. But we 
know that the law of gravitation requires that the areal 
velocity shall be a constant ; that is, the radius vector 
describes equal areas, instead of equal angles, in equal 

times (Art. 88). In one case — is constant; in the 

other, A r 2 — is constant. A little consideration will 

2 dt 

show that the mean value of r cannot be the same in 
the two cases. 

76. If y=f(x) and all of the dx's are equal, the 
average length of y between x = a and x = b is at once 

found by dividing the area I f(x)dx by b — a; for 

returning to Fig. 8, if the area MPSN be divided by 
its base b — a, the quotient is the altitude of an equiva- 
lent rectangle of base b — a] and the altitude of the 
rectangle is the average altitude of the strips repre- 
sented by TQRV\ that is, of the ys. 

Examples. 

1. Find the average length of the ordinates of a 
semicircle, supposing the series taken equidistant. 
We have x 2 + y 2 = r 2 ; or, y = Vr 2 — x 2 ; therefore 

M= — I Vr 2 — x 2 dx =+7rr. 

From this result it appears that the average ordinate 
equals the length of an arc of 45°. 



90 CALCULUS. 

2. Find the average length of the ordinates, sup- 
posing they are drawn through equidistant points on 
the circumference. 

In this case 

2r 



1 C n . 

= — I r si 



M=- rsinddO 



IT 



3. Given pv = c ; show that the mean pressure 

between the volumes v x and v» is log- 2 , v chang- 

ing by equal increments. 211 

4. A particle has simple harmonic motion. Find its 
mean velocity as it passes from the extremity of the 
radius to the center of the circle. 

77. The above geometric conception of mean values 
may be adopted when a function is expressed in polar 
coordinates. 

If r =/(#), let x be written for 6, and y for r, so 
that we have y=f(x). This equation furnishes a 
curve which sustains peculiar relations to the original 
polar curve. The radii vectores lose their fan-shaped 
arrangement, and are placed parallel and equidistant 
(if 6 is an equicrescent variable) with their extremities 
011 a common line, the :r-axis. The pole may be viewed 
as developing into this axis, — just as if a draw-string 
were let out, — while a circle of unit radius with the 
pole as center develops into a straight line parallel to 
the #-axis, the radii vectores keeping their position 
of perpendicularity with respect to the circumference of 
the circle. The mean value of the radius vector then 

1 r b 

becomes I f(x) dx, as before. 

b — aJ* 



WOBK. 91 

For example, to find r , the mean length of the radius 

a (\ _ e 2\ 

vector of the ellipse r = — - '-* 6 being an equi- 

F 1 + e cos 6 H 

crescent variable, we have, using one-half of the ellipse, 

1 r«a(l - e 2 ) 7 r A o 

r = — I — ^ —ax = a VI — e*. 

irJol + e cos # 

The radii vectores, now in the role of ordinates, are 
distributed at equal intervals through an area A whose 
base is 7r. 

Example. Find the average length of the radius 
vector of the cardioid r = a(l — cos #). 



M=- \ a (1 — cos #) cfe = — 

7T«yo 7T 



smo; 



= a. 



78. Work. If a force _F acts on a body of mass m, 
giving it an acceleration — -, 

XT d 2 s 

F=m dT* 

Multiplying by ds, 

Fds = m — f — )ds. 
dt\dtj 

Integrating between the limits v and V, V being the 
velocity when s = 0, and v the velocity when s = s, 

If F= 0, fFds=±mv 2 . 



92 CALCULUS. 

Fds is defined as the work done on the body m as it 

is moved through the space ds. I Fds is the work 

done in moving the body over the arc s. 

J mv 2 is defined as the kinetic energy which the body 
possesses because work has been expended upon it, the 
kinetic energy representing the work stored up in the 
body. In order to perform the operation indicated by 

I Fds we of course need to know what function F is 

of s in case F is a variable depending on s. Suppose 
that F=<p(s') and W represents the work ; then 



W 






From this it appears that work can be represented by 
an area referred to a space axis (x-axis), and a force 
axis (y-axis). 

79. In fact, the integral for area is seen to be repre- 
sentative of all definite single integrals, these integrals 

taking the general form I f(x)dx. The primary or 

horizontal axis is named for the quantity which x de- 
notes, and the secondary or vertical axis is named for 
the function of x. The integral itself is then repre- 
sented by the area MPSN. (Fig. 8.) 

80. Lengths of curves. Referring to Fig. 2, Art. 38, 
it is seen that 



ds = Vcte 2 + dy* =\1+ f^ 2 dx =yjl + (^ffdyi 



LENGTHS OF CURVES. 93 

hence, if s is the length of an arc from the point (x 1 ', ?/) 
to the point (x n , y n ), 



Convenience must decide which of the formulas we 
shall use in any given example. 

Examples. 

c f - 
1. Find the length of the catenary y = -I ec + e 

the curve in which a uniform chain hangs. 

dy_li 



dx 2 x€ 



therefore ^l+l-^-j = -(e c + e 

and s = I -le c + e c \dx = -le { 

2. Find the circumference of the circle x 2 + y 2 = r 2 . 

We have -f- = — - ; then, if AB is the first quad- 
dx y 

rantal arc of the circle, 



AB =SS l+ (t) 2dx ^ r \ i + 



dXJ *^0 \ y2 

dx 



^X~\ V ITT 

= r\ sin - = 

L ' Jo 

Therefore the circumference of the circle, 4 .Ai?, = 2 7rr. 



94 CALCULUS. 

81. Volumes of revolution ; areas of surfaces of 
revolution. 

If a plane curve revolves around any line in its plane 
as an axis, it is evident that tlie figure generated is 
such that any cross-section of it by a plane at right 
angles to the axis is a circle. The volume may be found 
b)^ taking the axis of revolution as the #-axis and add- 
ing together layers dx in thickness. The area of any 
cross-section is Try 2 . If V represents volume, we have 

then „ b 

V= I Try 2 dx, 

in which the equation to the generating curve is 

v =/(■>■)■ 

Similarly, the surface may be found by noticing that 
the arc 8s generates the frustum of a cone whose sur- 
face is known from elementary geometry to be 

or, in the limit, 2iryJs\ so that, if S is the area of the 

surface, / , 7 N0 

dy\ 2 



H>W 1+ @D* 



Examples. 

0-2 n,2 



1. The ellipse ^- + f- = l revolves about its major 
axis. What is the volume generated ? 

1 V= ( iry 2 dx = ( it — (a 2 — x 2 ~) dx = — - I (a 2 — £ 2 ) dx 

Jo Jo a 1 a 1 Jo 



irb 2 [ 2 



VOLUMES AND SURFACES. 



95 



The entire volume is therefore |- irab 2 . 
The volume of the sphere, ^ 7m 3 , is a special case, in 
which b = a. 

2. Find the area of the surface generated as the 
ellipse revolves about its major axis. 



%s 






+ffi* 



2irb 



— CV&-(a 2 -b 2 )x 2 fdx 

a 2 Jo 



= irb\ b + 



w 



^J a 2 — b 2 
therefore the whole surface is 

w 



• _iVa 2 — b 2 ' 

sin 

a 



2irb 
= 2irb 



b + 



b + 



. _iVa 2 — b 2 

_sm 

/ a 2 _ ^2 a 



a 2 1 6" 

— — — cos - - 

Va 2 - b 2 a. 



3. Find the area of the surface of a sphere whose 
radius is a. 

If we make b = a, we have, from the result in the 
preceding example, for the area of the surface of the 
sphere, 



2 ira 



a + a 2 



cos - - 
a 



~Va 2 — a 2 _ 
We must now find the value of 



2ira 



a + a 2 



) 



Va 2 — b 2 



when b = a. 



96 



CALCULUS. 



Treating b as a variable and applying the principle 
of Art. 32, 

1 



b - 

COS - - 

a 


,4( C0S ~ 

,7 


■31 


Va 2 - b*J 


b=a — Va 2 - 
db 


w> 2 



^-5 n i 



b 



Va 2 -6 2 



Therefore, $ = 2 7ra ( a + a 2 - ) = 4 ira 2 . 



This result for the area of the surface of a sphere 
agrees, of course, with the one obtained by the method 
of elementary geometry. 

82. The area integral I f(x) dx represents the sum 

%sa 

of strips whose height is y and breadth dx. We may 
reach the same result by starting with the elementary 
rectangle dxdy and using two integral signs, — one to 
indicate that we add such rectangles together to make 
a strip y in height, and a second to indicate that the 
strips are to be added together, making the area from 
a to h (Fig. 8). For example, the area of the ellipse 
may be found by adding together the areas dxdy from 
the major axis to the curve itself ; then adding together 
the strips from the minor axis to the end of the major 
axis. To indicate this double operation, we write 



- I I dxdy, 



using the right-hand integral sign with dy. 



DOUBLE INTEGRALS. 



97 



Performing the first operation, 

I dxdy = I ydi 

•/} *^0 



The remaining part of the work is the same as in 
Art. 74, example 2. 

The above procedure in finding areas involves what 
is known as a double integral. Similarly, three succes- 
sive indicated integrations constitute a triple integral. 
Examples of double integrals will occur in subsequent 
articles. 

83.* Suppose a point to travel once round the closed 
oval area J., an indicator diagram, for instance, so as 
always to have the interior of the curve on the left 




Fig. 9 



hand. Let B be the minimum point, and C the maxi- 
mum point with respect to the #-axis ; D the minimum 
point, and E the maximum point with respect to the 
^/-axis. 

* Greenhill's Differential and Integral Calculus. 



98 CALCULUS. 

Then A = ( J dxdy = I xdy, 

taken round the perimeter of the curve. 

From B to O along BPC, dy is positive, and 

fxdy = area MBP ON. 

From (7 to B along C(?Z?, tf ?/ is negative, and 

j*xdy = - area 3IBQCJST; 

so that, taken round the curve, 

fxdy = area MBPCN- area 3IBQCN= A, 

the area of the closed curve. 

But 1 1 dxdy = ( ycfo ; 

and from E to D along EPD, dx is negative, so that 

J^/.r = - area LEQJDK; 

and from i) to JE along DBE, dx is positive, so that 

fydx = area LBBBK; 

and therefore, taken round the curve, 

I ydx = — A. 

Therefore taken round the curve, 

j (ydx + xdy) = ; 

and ydx + xdy = d (xy) is called a perfect differential. 
Its integral between two limits is independent of the 
intermediate values of x and y and of the path described 



MOMENT OF INERTIA. 99 

between the limits; so that, taken round any closed 
path, the integral is zero. 

When Fig. 9 represents an indicator diagram, and 
KL the reduced stroke of the piston, while the ordinate 
y represents the pressure of the steam, the pencil will 
describe the contour with the area to the left, when the 
steam pressure is urging the piston from L to K. The 
diagram taken on the return stroke from the other end 
of the cylinder will be described in the opposite sense, 
with the area on the right hand of the describing pencil. 

84. Moment of inertia. When a rigid body rotates 
about an axis, the linear velocity of any particle of the 

boc ^ is ds rae 

v = — = = r&), 

dt dt 

a) being the angular velocity of the particle, and r its 
distance from the axis. Its kinetic energy of rotation is 
therefore ^ mv 2 = ^ mr 2 co 2 , m being the mass of the par- 
ticle ; and the kinetic energy of rotation of the whole 

body is 

1 mv i _|_ i m V 2 + | m n v ,!2 + ••• 

= 2 ^ mv 2 = 2 ^ mco 2 r 2 = ^ ft) 2 Smr 2 ; 

that is, one-half the product of the square of the angular 
velocity and 2mr 2 . 

The symbol 2 is used to indicate a pol} r nomial in 
which the terms are similarly constituted, as in the case 
before us. Since such an expression as 2 J mv 2 is in 
reality a polynomial, only common factors can be re- 
moved and placed before 2, the symbol of summation. 
Thus in 2 \ mco 2 r 2 , \ is, of course, a common factor ; co 2 is a 
common factor because the rotating body is supposed to 

LofC. 



100 



CALCULUS. 



be rigid, and consequently all of its parts have the same 
angular velocity ; but m is not a common factor because 
it is not supposed that all of the particles have equal 
masses ; neither is r a common factor, for the particles 
are at different distances from the axis of rotation. 

The quantity 2 mr 2 is called the moment of inertia of 
the body with respect to the axis, and is seen to be the 
sum of the products obtained by multiplying the mass of 
each particle by the square of its distance from the axis. 

If a body rotates with a given angular velocity about 
different axes, the kinetic energy of rotation with respect 
to any axis must be proportional to 2 mr 2 ; consequently, 
the moment of inertia measures the capacity of a body 
to store up kinetic energy during rotation about the axis 
with respect to which the moment of inertia is taken. 



Examples. 

85. l. A sheet of metal, rectangular in shape and 
of uniform density, is made to rotate about an axis 

coinciding with one end. 
What is its moment of 
inertia ? 

Take the axis of rota- 
tion for the #-axis with 
the origin at the left-hand 
corner of the rectangle. 
Let b be the breadth and 
-2 % d the height of the rec- 
tangle. If p is the density 
of the metal, pdydx is 
the mass of the indefi- 
nitely small rectangle dy dx cut anywhere from the 



i 



Fig. 10. 



MOMENT OF INERTIA. 101 

sheet ; and (pdydx^y 2 is the moment of inertia of 
this small piece. Hence the moment of inertia of the 
entire sheet becomes 

i Jo ^ f ^ = al yUydx 

= phj*fcly = f*f. 

From this example it is plain that in all cases in which 
the density is constant throughout the body, the density 
factor may as well be set aside until the integration is 
completed. If, however, the density varies from point 
to point, so that p is some specified function of x and y, 
it must be kept under the sign of integration and be 
taken account of in the process of integrating. 

2. A straight slender rod of length Z, whose density 
varies directly as the distance from one end, rotates 
about an axis perpendicular to it and passing through 
the end having the least density. What is the moment 
of inertia with reference to this axis ? 

Take the given axis as the #-axis, with the origin at 
the end of the rod. 

pccy ; therefore p = ky if Ar is the density at a unit's 
distance from the end. Then the moment of inertia is 



£py 2 dy = JkyHy = — • 



3. Find the moment of inertia of a circle with refer- 
ence to an axis through its center and perpendicular to 
it, p being a constant. 



102 CALCULUS. 

Let R be the radius of the circle, r the distance of 
any particle from the axis, and 6 the variable angle 
measured from some chosen radius. Consider an ele- 
mentary portion bounded by the circles whose radii are 
r and r + dr, and by the radii forming the angle d0. 
In the limit this bit of area becomes the rectangle 
(rd0)dr; hence, the integral is 

C ' C \\rdrd6) = 2tt Crhlr=-W; 

and therefore the moment of inertia is ™ 

2 

86. Kepler's laws. It is shown in works on the 
determination of orbits* that the equations for the un- 
disturbed motion of a planet or comet relative to the 
sun are : 

g + F(l + ™)£ = 0, (1) 

g + F(l + m)5 = 0, (2) 

g + *. ( l + Bl )* = 0, (3) 

in which x, y, z are the coordinates of the heavenly 

body referred to the sun as origin, — -, — f, — - are 

dt l dt l dt z 

the accelerations parallel to the three axes of reference, 
r is the distance of the body from the sun, k 2 is the mass 
of the sun, and m the ratio of the mass of the body to 
the mass of the sun. Having these three equations, we 

* Watson, Theoretical Astronomy; Dziobek, Planeten-Bewe- 
gungen ; Tisserand, Determination des Orbites. 



KEPLER'S LAWS. 103 

can at once establish Kepler's laws. Arts. 87, 88, 89, 
90, 93 are taken, with slight changes, from Watson's 
Theoretical Astronomy. 

87. If we multiply equation (1) of the preceding 
article by y, and equation (2) by x, and subtract the 
last product from the first, we shall have, after inte- 
grating the result, 

xdy — ydx _ n 
Jt =C ' 

C being the constant of integration. 
In a similar manner we obtain 

xdz — zdx _ pi ydz — zdy _ „,, 
dt dt 

If we multiply these three equations respectively by 
2, — y, and #, and add the products, 

Cz-C f y + C"x=0. 

This is the equation to a plane passing through the 
origin of coordinates (Art. 142). Since x, y, z are the 
coordinates of the heavenly body, it must remain in this 
plane. The path of the heavenly body relative to the 
sun is therefore a plane curve, and the plane of the orbit 
passes through the center of the sun. 

88. If we multiply equations (1), (2), and (3) re- 
spectively by 2 dx, 2 dy, and 2 <fe, take the sum and 
integrate, we have 

dx 2 + dy 2 + dz 2 | y&Q \ m ) C xdx + y d y + zdz = p. ^ 



104 CALCULUS. 

But r 2 = x 2 + y 2 + z\ 

therefore rdr = xdx + ydy + zdz. 

Introducing this value of xdx + ?/^/ + zdz into equa- 
tion (4) and performing the integration indicated, we 
have 

^ r +/ '"°' (5) 

h being the constant of integration. 

If we add together the squares of the expressions for 
C, C, 0", and put C 2 + C" 2 + C" 2 = 4/ 2 , we shall have 

(x 2 + y 2 + z 2 ) (d.r 2 + di/ 2 + ch 2 ) _ (xdx + ydy + zdz) 2 = ifi . 
dt 2 dt 2 f ' 

,, , . o dx 2 + d// 2 + dz 2 r 2 dr 2 , /., , a ~. 

that is, r 2 - JL. ____ = 4/ 2. (6 ) 

If we now represent by dv the infinitely small angle 
contained between two consecutive radii-vectores r and 
r + dr, since dx 2 + dy 2 + dz 2 is the square of ds, the ele- 
ment of path described by the body, we shall have 

dx 2 + dy 2 + dz 2 = dr 2 + r 2 dv 2 . 

Substituting this value of dx 2 + dy 2 + dz 2 in equa- 
tion (6), 

r 2 dv=2fdt. (7) 

The quantity r 2 dv is double the area included by the 
element of path described in the element of time dt, 
and by the radii-vectores r and r + dr. See Fig. 4, 
Art. 46. The area of the triangle POP' = area POQ 
+ area PQP 1 ; but in the limit area PQP f vanishes, 



KEPLER'S LAWS. 105 

and area POQ=\OP(PQ)= | r (n20) . W riting equa- 
tion (7) in the form 

\fdv_ f 
dt 7 ' 

1 r 2 c J v 

the quantity -2— — is the area described by the radius- 
vector in the time dt, divided by the time, and is defined 
as the areal velocity. Since/ is a constant, we conclude 
that the radius-vector of a planet or comet describes equal 
areas in equal intervals of time. (Kepler's second law.) 



89. Combining equations (5) and (6) so as to elimi- 
nate ^ , and solving for dt, we have 

dt 2 

dt = ™^ (8) 

V2 rk 2 (1 + m) - hr 2 - 4 J* 

Substituting this value of dt in equation (7), 

dr _ r V2 rk 2 (1 + ni) — hr 2 — 4 f 2 
dv~ 2/ 



(9) 



We have seen (Art. 46) that the condition that r 

shall be a maximum or minimum is = 0. With 

rd9 

the notation of the present article, v = 6 + some con- 
stant ; therefore dv = dd, and we have, in order to find 
the maximum and minimum values of r, 



V2 rk 2 (l + m)- lir 2 - -if 2 = Q . 

2/ 

that is, 2 rk 2 (1 + m) - hr 2 - 4 f 2 = 0. 



106 CALCULUS. 

If 1\ and r 2 represent the two roots of this quadratic 
equation, 

_ F(l + m) / 4/» ^(lT^7 



_ F(l + w) J 4 p /H (1+w ) 2 

Since the equation of condition yields only two values 
of r, the orbit cannot have more than two apsidal points. 
If it is a closed curve and not a circle, it must evidently 
have two, rather than one, such points. The point 
corresponding to r v the maximum value of r, is called 
the aphelion, and the point corresponding to r 2 is the 
perihelion. 

90. If we put 



ga+ia + y-4g + * i p+-y. < i +0 



h * it A 2 

and add the two expressions, we have 

a 

Also, taking the difference of the two expressions and 
substituting the value of h just found, 

4/ 2 = ak\\ + m)(l - e 2 ) = *»p(l + m) 

if p be written for a(l — e 2 ). 



KEPLER'S LAWS. 107 

Substituting these values of h and 4/ 2 in equation (9) 
it becomes 



7 '\f~vdr 

av 



id 



i 



^r-l^-p \ll-( l A 



a ■* * \e r e) 



the integral of which gives 



v = cos ] -(^ — 1 ) + G), 
co being the constant of integration; and, therefore, 
we have -f - — 1 ) = cos (# — ©); 

that is, solving for r, 

r = , ^ r (10) 

1+6 cos (v — CO) v y 

This expression is seen to be the polar equation to a 
conic section (Art. 115), the pole being at the focus, p 
being the semi-latus rectum, e the eccentricity, and co 
the angle at the focus between the major axis and a 
fixed line in the plane of the orbit. #, the vectorial angle, 
is measured from this latter line. 

If co = 0, equation (10) becomes 

r = —? = i a(1 -* 2) - (11) 

1 + e cos v 1 + e cos v 

In this case v is called the true anomaly. We now 
conclude that the orbit of a heavenly body revolving 
around the sun is a conic section with the sun in one of 
the foci. (Kepler's first law.) 



108 CALCULUS. 

91. The planets revolve around the sun in ellipses, 
and these ellipses are, as a rule, characterized by small 
eccentricities. Thus the eccentricity of the earth's orbit 
is at present 0.01G77. Of all the major planets Mercury 
has the most elliptic orbit, its eccentricity being 0.2056. 

The orbits of comets, on the other hand, may be 
described as parabolic, by which we mean that they are 
either ellipses of great eccentricity (almost unity), or 
hyperbolas whose eccentricity differs but little from 
unity. In many cases the eccentricity cannot be found 
to differ from unity ; the orbit is then of course described 
as a parabola. Of the periodic comets which have been 
observed at more than one perihelion passage, Tempel's 
comet has the least eccentricity, namely : 0.4051. 

92. In putting r x and r 2 equal to a(l + e) and a(l — e) 
respectively, the argument has much the air of begging 
the question, seeming to assume that the orbit is a conic 
section and then using the assumption in the proof. 
But it is to be noted that when we adopt the expres- 
sions a(l + <?) and a(\ — e), e does not mean eccentri- 
city, neither does a mean semi-major axis. They do not 
bear these meanings until in equation (10) we identify 
them with the constants in the polar equation of 
coordinate geometry. 

93. If the values of h and 4/ 2 , as found above, are 
introduced into equation (8), we have 



7 , Va rdr 

at-. 



k Vl + m VaV — (a — r)' 2 



KEPLER S LAWS. 
which may be written 



109 



dt = — 



ae 



&V1 



m 



VW^Y 



ae 



or 



dt 



kVl + '. 



-d 



a — r 

ae 



a — r -.fa — r 
a 



Vi- 



a — r 
ae 



+ e 



ae 



ae 



v» 



— r\2. 



a — r 

ae 



the integration of which gives 

3 

a 2 



t = 



&Vl + 



m 



cos" 



-«Jl- 



ae 



ae 



+ C. (12) 



When the heavenly body is in perihelion, r — a(l — e), 
and the integral reduces to t' = 0; therefore, if we 
denote the time from perihelion by t , we have 



t n = 



*Vi~T 



ill 



COS" 



a — i 
ae 



-e\l 



a — r\- 
ae 



(13) 



We here integrate between the limits t ! and £, with 
t — t' = t . 

In aphelion r = a (1 + e) ; putting this value of r 
into equation (13), and denoting by \t the time from 
perihelion to aphelion, we have 



1 t — 

2 T ~ 



&Vl + 



7T. 



(14) 



W2 



According to Kepler's second law, the time from 
aphelion to perihelion must equal the time from peri- 



110 CALCULUS. 

helion to aphelion ; therefore r is the time of a com- 
plete revolution. 

The time of a complete revolution is termed the 
periodic time. 

From equation (14) 

t 2 = 4tt 2 i aS : (15) 

& 2 (l + m) v J 

and for a second planet, 

t' 2 = 4tt 2 t> f' S t . (16) 

A 2 (1 + m f ) V y 

Comparing equations (15) and (16), we see that 

(1 + 111) T 2 = a 3 ,.rj, 

(l + m')r' 2 a' 8 " ^ ' 

If the masses of the two planets are very nearly the 
same, we may take 1 + m = 1 + ?n' ; and hence, in this 
case, it follows that the squares of the periodic times of 
tivo planets are to each other as the cubes of the semi- 
major axes. (Kepler's third law.) 



CHAPTER IV. 

ANALYTIC GEOMETRY. 

94. In this chapter it is proposed to present the 
elementary principles of analytic (coordinate) geometry, 
with especial reference to conic sections. 

The Cartesian system* of coordinates has already 
been explained in Art. 36. Here, as there, we shall 
speak of the curve F(x, y) = 0, the curve y =f(x'), the 
line ax + by + c = 0, etc., instead of saying " the curve 
which the equation F(x, y) = represents," etc. 

95. If the equations 

y = 0O), (6) 

are treated as simultaneous, the x and y of equation (a) 
must mean the same as the x and y of equation (6); 
consequently, as coordinates they are restricted to the 
point or points common to the two curves (a) and (5). 
If equations (a) and (6) have been so combined as 
to eliminate one of the coordinates, say y, the x of the 
resulting equation is the abscissa of the point of inter- 
section of the two curves, and the curves intersect in 
as many real points as there are real roots of this new 
equation. 

* Called the "Cartesian system," after Rene' Descartes (1596- 
1650), the inventor of coordinate geometry. 

Ill 



112 CALCULUS. 

For example, if y be eliminated between the two 
equations x 2 + y 2 — 4 = and x + y — 1 = 0, we have 
x 2 — x — f = 0. \ ±\ V7 are therefore the abscissas of 
the points where x 2 + y 2 — 4 = and x + y — 1 = 
intersect. 

96. If, however, equations (a) and (5) have been so 
combined that neither x nor y is eliminated, the x and ^/ 
now refer primarily to the points common to the curves 
of (a) and (?>) ; but we may treat them as a new x and 
y, — the current coordinates of a point describing a 
new curve, which passes through the intersections of 
the curves (a) and (J). 

For example, suppose we have the equations y=2x — 2 
and 2y = x + 2. Adding them, y = x, a straight line 
distinct from the given lines, but passing through their 
point of intersection. 

97. If the coordinates of a given point satisfy a 
given equation, the point evidently lies on the curve 
which the equation represents. Conversely, if a point 
is on a curve, its coordinates will satisfy the equation 
to the curve. 

If an equation F(x, y) = can be written 

f(x, y)<£(>, y)=o, 

the curve of the given equation is made up of the com- 
bined curves of f(x, y} = and $(x, y)= ; for any 
point whose coordinates cause m f(x, y) to vanish, thus 
satisfying the equation f(x,y')=0, will also cause 
/(#, y}<\>(x, y} to vanish. Hence, all points 011/(2;, y^} = 
are also points on f(x, y^)$(x, y) = 0. Similarly, all 
points on $(2;, #)= are points on f(x, y^)$(x, ^/)= 0. 



ANALYTIC GEOMETRY. 113 

Further, there are no other points on f(x, y^)c\>(x, y) = 0, 
because f(x, ^)</>(#, y) cannot vanish except by the 
vanishing of either f(x, y) or <£(#, y). 
For example, 

x 2 + 2 y 2 + 3 xy — x — y = (x + y) (x + 2 y — 1) ; 

hence, the curve which x 2 + 2 y 2 + 3 xy — x — y = 
represents is made up of the straight lines x + y = 
and x + 2y — 1 = 0. 

(The term " curve " is here used as inclusive of 
straight lines.) 

98. If we have the equations y =f(x) and y = <£(#), 
the equation y =f(x)<f>(x) represents a curve whose 
ordinate for any abscissa x ! is the product of the orcli- 
nates corresponding to x' in the two primary curves. 

For example, if y = x and y = log x, y = x log x is 
a third curve whose ordinate at any point equals the 
product of the corresponding ordinates. 

In drawing such a set of curves to the same axes of reference 
it is well to use colored pencils or crayons. For instance, if the 
straight line y = x is drawn in red, the logarithmic curve in yellow, 
and the curve y = x log x in blue, the resulting diagram appeals to 
the eye much more forcibly than if all were done in black or white. 

99. If y =f(x) and y = <\>(x), the equation 

represents a curve whose ordinate at any point is the 

sum of the corresponding ordinates of the given curves. 

For example, the so-called " equation of time" is 

made up of two parts : one due to the eccentricity of 



114 CALCULUS. 

the earth's orbit, the other to the obliquity of the 
ecliptic. If E x and E 2 represent these two parts re- 
spectively, E, the whole equation of time, equals E x + E v 
With a scale of dates one year long for the x-axis, and 
a scale marked to minutes for the ?/-axis, we may con- 
struct the curve of E 1 and also the curve of E v A 
third .curve, whose ordinate for any date is the sum of 
the ordinates of the first two curves for that date, then 
represents E. See Young's General Astronomy, Fig. 64, 
edition of 1898. 

The principles of this section and the preceding one 

f(x) 

can evidently be extended to such forms as y = ^ , 

y =/(*)- <K*)> et€. * w 

100. If we move the #-axis parallel to itself through 
the distance y\ every ordinate is changed by the amount 
y' . Similarly, if the y-axis is moved parallel to itself 
through the distance x', every abscissa is changed by 
the amount x 1 '. So, if X and Y are the new current 
coordinates, x = X + x' and y = Y + y\ in which x r and 
y' are the coordinates of the new origin referred to the 
old axes. Hence, if in any equation F(x, y) = 0, we 
write x + x' for x, and y + y' for y, so that the equation 
becomes F(x + x', y + y r ) = 0, the geometric result se- 
cured is a change of origin to a new point (x } ', y f ), with 
new axes parallel to the old ones. 

The new current coordinates may be written x, y, 
instead of X, I 7 , since they do not occur in connection 
with the old coordinates and therefore cannot be con- 
fused with them. 

For example, x 2 + y 2 = r 2 being the equation to a circle 
with its center at the origin, (x — a) 2 + (y — b) 2 = r 2 is 



ANALYTIC GEOMETRY, 



115 



the same circle with its center at (a, 5). The coordi- 
nates of the new origin referred to the old axes are 
— a, — h. 

101. Suppose the axes to rotate around the origin 
through the angle «. The new coordinates of any 

point P are 

X= 0B f ; Y=PB f . 




Fig. 11 



Now 
OB 1 cosa = 0C=0B + BC=x + BC=z + PB f sin «. 

Therefore x = OB' cos a — PB f sin « 

that is, x — X cos a — Y sin a. 

Similarly, ^ = Xsin a -f- 1" cos a, 



116 CALCULUS. 

Hence, if in any equation F(x, y)=0 we write 
x cos a — y sin a for x and # sin a + y cos a for y, the 
geometric result is the rotation of the axes through 
the angle a, in which a may have any value and be 
positive or negative. 

102. In the formulas just derived, the old coordi- 
nates x and y are explicit functions of the new coordi- 
nates X and Y. If we multiply the first formula by 
cos a and the second by sin a, and add the products, we 
have 

X = x cos a + y sin a. 

Similarly, Y — — x sin a -f- y cos a. 

The new coordinates are now explicit functions of 
the old ones. 

103. The formulas derived in the three preceding 
articles are indispensable in astronomy. As an 
example of the use of the two in Art. 102, suppose a 
planet is referred to the line in which the plane of its 
orbit cuts the ecliptic as the a>axis, with a line at right 
angles to it in the ecliptic as the ?/-axis. The planet 
may be referred to a new #-axis having its positive end 
directed toward the vernal equinox, with a correspond- 
ing new ?/-axis, if we use the relations 

X = x cos Q> — y sin Q> , 
Y — x sin Q> + y cos Q> . 

The axes are here moved backward, that is, in the 
negative direction, through the angle &> (the longitude 



ANALYTIC GEOMETRY. 117 

of the ascending node), Q> being the angle between the 
vernal end of the equinoctial line and the line passing 
through the sun and the point through which the 
planet moves in going from the south to the north side 
of the ecliptic. 

104. We have seen (Art. 39) that ax + by + c = 

represents a straight line because -£ = , a constant. 

dx b 

It follows that if in any two equations 



y = mx + ?i, 
y = m ! x + n f , 

m ! = m, the lines are parallel, for they have the same 

slope. 

Also, if m 1 = , the lines are at right angles to 

m 

each other ; for m and m' are now the tangents of 
a and 90° + a ; since tan (90° + «) = — cot a. 

105. If x is put equal to zero in the equation F(x, y) = 0, 
the resulting value of y must be the ordinate of the 
point where the curve crosses the y-axis. Now if x = 

in ax + by + c = 0, y = ; but — - is the constant 

b b 

term when the equation is written in the form 

a g 
y = — - x — -. It follows that if any linear equation 
b b 

be written in the form y = mx + n, the constant term is 
the distance from the origin to the point where the line 

crosses the y-axis. 



118 



CALCULUS. 



106. Suppose that P 1 '(V, ?/), P" (x" , y ,f } are any two 
points on a straight line, while P(x, y) is the moving 

point. From similar 
triangles (Fig. 12), 

PQ r _P"D. 




F'Q' 


PD' 


that is, 




y - >/' 


_y"-y' 


x — x' 


x" — x' 



This equation we 
describe as the equa- 
tion to the straight 
line in terms of the coordinates of two points through 
which it passes. 

For example, the points (—2, 1), (3, — 4) determine 
a line whose equation is 



y 



1 



4-1 



z-(-2) 3-(-2y 

which becomes, after reducing, 

x + y + 1 = 0. 

If P ! and P n are indefinitely near each other, 
y n — y f becomes dy, and x n — x' becomes dx. Hence 

y - y f = ^f^ - ^)' 

which is the equation to the straight line in terms of 
its gradient (or slope) and one point through which it 
passes. 



ANALYTIC GEOMETRY. 119 

107, Referring to Fig. 12, Ave see that if D is the 
distance between any two points, 



j) = v<y ' - x r y 2 + <y - ^) 2 - 

Also, if P n is midway between P and P', 

*" = 10 + *0 and y" = Ky + y') 5 

that is, the coordinates of a point bisecting the distance 
between two points are the averages of the abscissas 
and of the ordinates, respectively, of the given points. 

108. To find the perpendicular distance from a given 
point (V, 2/) to a given line 

y = mx + n, (1) 

we write y — y f = m(x — #') ; (2) 

a line passing through (V, y f } parallel to (1). The 
intercept of (1) on the y-axis is w, the intercept of (2) 
on the y-axis is y r — mx 1 ', and the difference of these 
intercepts is ^ — m#' — n. It is evident that if this 
difference of intercepts be multiplied by cos (tan -1 ???)«> 

that is, by — , we shall have the perpendicular 

Vl + m 2 
distance between the two lines, and hence the distance 
from (V ', y f ) to y = w?^ + w. The formula for the dis- 
tance from any point (V, y f ) to any line y = mx + n is 

therefore 

y — W22/ — w 

Vl + m 2 
Or, if the equation to the line be in the form 
ax + by + c = 0, 



120 

the formula is 


CALCULUS. 
r , a f , c 


which equals 


ax f + by ! + c 



Va 2 + b 2 

It appears, then, that we have simply to evaluate the 
function ax + by + c for the coordinates of the given 
point and divide by the square root of the sum of the 
squares of the coefficients of x and y. 

Exercises. 

1. Given the equation ax + by + c = 0, show that if it 
be written in the form 

c c 



the quantities standing beneath x and y are the intercepts 
on the axes of x and y respectively. 

2. Write the formula for the distance from the origin to 
the line ax + by + c = 0. 

3. Find the equation to a straight line which passes 
through a given point (_p, g) and makes equal angles with 
the axes. 

4. Find the length of the perpendicular from the origin 
on the line a (x — a) + b (y — b) = 0. Also, find the portion 
of this line intercepted by the axes. 



ANALYTIC GEOMETRY. 121 

5. Write the equation to a line passing through 
the origin and making an angle of 120° with the 
a>axis. 

6. The coordinates of the vertices of a triangle are 
(1, 2), (—3, i) 3 (4, a/2); write the equations to its 
sides. 

7. Find the equation to a line which passes through 
the intersection of the lines x = a, x +y + a = 0, and 
through the origin. 

8. Show that the lines y = 2 x -{■ 3, y = 3 x + 4, 
y = 4 # + 5 all pass through one point. 

9. Find the slope of the line y = mx + 3 in order 
that it may pass through the intersection of the lines 
y = x + 1 and y = 2 x + 2. 

10. Find the equation to the straight line which is 
equidistant from the two lines y = mx + n ± ?i f . 

11. If a is the angle between the lines y = mx+n 

and y = m'x + n ! , show that tan a = — — 

1 + wim' 

109. The ellipse. Suppose that the circle x 2 + y 2 = r 2 
is in a plane JfeT which makes an angle 7 with another 
plane If. Let the #-axis to which the circle is referred 
be parallel to plane If. If perpendiculars to jV are 
dropped from the extremities of the ordinates of the 
circle, the feet of these perpendiculars will form in If 
a new curve which is the projection of the circle on 
the plane If; and any ordinate y f of the circle be- 
comes the ordinate y f cosy in the new curve. Hence, if 
we write the circle-equation in the form y = ±Va 2 — z 2 , 



122 CALCULUS. 

the new curve, called the ellipse, has for its 
equation 

y = ± cos 7 Va 2 — x 2 . 

a cos 7 is a constant, and is evidently that line of the 
ellipse which replaces that radius of the circle which is 
at right angles to the #-axis. Put acosy = b; then 

cos 7 = -, and the ellipse-equation becomes 



y — ±- Va 2 



~2. 



a 



.2 2 

that is, — + ^. = 1. 

a 1 b z 

a is called the semi-major axis, and b the semi-minor 
axis of the ellipse. 

The student should distinguish carefully between the 
axis of reference (a?-axis) and the major axis. An axis 
of reference is a mere convenience, and we might study 
the ellipse with such an axis occupying some other posi- 
tion in relation to the curve, or even without any such 
axis ; but the major axis is an essential line of the 
ellipse, occupying a special position within it. The 
same is true of the minor axis. 

110. Now let the ellipse and the circle be drawn in 
the same plane, — the ellipse inside the circle with the 
major axis coinciding with that diameter of the circle 
which lies in the #-axis. 

The ellipse has the appearance of a circle flattened in 
the direction YY f . AA! = 2 a is the major axis, and 
BB ! = 26 is the minor axis. A and A 1 \ the extremi- 
ties of the major axis, are called the vertices. 



ANALYTIC GEOMETRY 



123 



With jB, one extremity of the minor axis, as a center 
and with a radius equal to a, strike an arc. This arc 
will cut AA! at points F and F' equally distant from 0, 
the common center of the circle and the ellipse. Evi- 
dently the flatter the ellipse is, the farther these points 
will be from ; hence, if we know the ratio of OF to 
OA, we know how flat the ellipse is, compared with the 
circumscribed circle. 



BF=a> £0 = b; therefore OF = Va 2 - 6 2 , and 



OF 
OA 



Va 2 — b 2 




Fig. 13. 



This important ratio is called the eccentricity of the 
ellipse, and is denoted by e. The somewhat similar 

(£ Q 

ratio, , is called the ellipticity of the ellipse. 

ct 

The eccentricity is evidently a proper fraction. The 
points F and F' are called foci ; and the double ordi- 
nate through either focus is known as the latus rectum. 



!4 


Since 


OJ 7 


C4Z,CDX?7,S. 




111. 


= V« 2 - 
= a(l 


- 6 s , 






Va 2 — 


F 


FA = a - Va 2 
Similarly, 


-6 2 



= a(l-e). 



From the way in which the points F and F f Avere 
found we see that the sum of the distances BF and BF ! 
is 2 a. It may now be shown that the sum of the dis- 
tances from any point P(x\ ?/) on the ellipse to the foci 
is 2 a. 

The coordinates of F, the right-hand focus, are ae, ; 
hence, by Art. 107, 

(FF) 2 = O' - aey + ij 2 

— x 12 — 2 aex 1 + a 2 e 2 + y' 2 . 

Since P is on the ellipse, its coordinates must satisfy 
the equation to the ellipse, and we have 

b 2 
y' 2 = — (a 2 — x 12 } 

= (l-e 2 )(a 2 -x f2 ). 
Substituting this value of y 12 in the expression for 

(Fpy, 

(FP) 2 = x ,2 -2 aex f + a 2 e 2 + (1 - e 2 )(a 2 - x /2 ) 
= a 2 — 2 aex f + e 2 x' 2 ; 
therefore FP = a — ex 1 . 

Notice that the other root, — (a— ea/), is rejected 
because a > ex 1 and FP is positive. Repeating the 



ANALYTIC GEOMETRY. 125 

argument for the distance F'P, the coordinates of F ! 
being — ae, 0, F'P = a + ex 1 ; hence FP + F f P = 2 a. 

112. The proposition just established affords a way 
of mechanically constructing an ellipse. Fasten one 
end of a string at a point F on the blackboard or paper, 
and the other at a point F', taking the distance FF f • 
somewhat less than the length of the string. Pass the 
string around a pencil and move the point of the pencil 
over the paper, keeping the string taut. An ellipse 
will be described. 

It is clear now that we might define an ellipse as the 
path of a point which moves so that the sum of its dis- 
tances from two fixed points is a constant. 

113. If b = a in — + *f- = 1, the equation returns to 

the circle-equation x 2 + y 2 = a 2 ; also, e = = 0. 

It thus appears that a circle is merely an ellipse with 
equal axes and eccentricity equal to zero. 



114. Let a line DP f (Fig. 14), be drawn parallel 
to the minor axis. If the equation to the ellipse is 

— + &- = 1, the minor axis lies in the ?/-axis, and 
a 2 o 2 

hence the line DD f is parallel to the ?/-axis. If its dis- 
tance from the axis is -, its equation is x = — , the 

e e 

equation affirming that whatever may be the ordinate 
of the point tracing the line, the abscissa is constantly 

-. Now the distance from any point P(x } ', y 1 ^) on the 
e 



126 



CALCULUS. 



curve to this line is - minus the distance of the point 
from the y-axis ; that is, - — x r , or 



a — ex 



We have 

already seen (Art. Ill) that the distance from P to the 
focus is a — ex 9 . Hence the distance from any point on 
the ellipse to the focus and the distance from the point 

to the line x = - are in the ratio 
e 

a — ex f 

a — ex' ' 



that is, e. Accordingly the ellipse may be defined as 
the path of a point which moves so that its distance from a 
given jived point and its distance from a given fixed line 
have a constant ratio less than unity. 

The line x = - is called the directrix. 

e 

115. In Fig. 14 let FP = r and angle EFP = 6. 



Y 






D 






P 


C 






( ° 




^ 


E X, 




L F J 










D' 



Fig. 14. 



ANALYTIC GEOMETRY. 127 



We have 


FF=--ae; 
e 




jPi = rcos(18O°-0) 




= — r cos 6, 


and also 


FP 
PC""' 


then FP = 


:ePC=e(FF+FL); 


that is, 


fa 

r = el ae — r cos 

\e 




= a — ae 2 — er cos i 


Solving for r, 


1 + e cos 



This is the equation to the ellipse in polar coordi- 
nates with the pole at the right-hand focus. From the 
point of view of astronomy it is the most important of 
all forms of ellipse-equations. See Art. 90. 

116. It is of interest, logically, to note that the 
theory of the ellipse may be developed from the defini- 
tion given in Art. 112, or the one in Art. Ill, or, 
indeed, from any fundamental property. In the present 
instance we have chosen to begin by viewing the ellipse 
as the projection of a circle on a plane making a given 
angle with the plane of the circle. 



-A 



ir 



117. The hyperbola. About the ellipse — + *- = 1 

circumscribe a rectangle with its sides parallel to the 
axes of the ellipse. Draw the diagonals of the rect- 



128 



CALCULUS. 



angle. Half of one of the diagonals is Va 2 + b 2 . 
With a radius of this length and with the center 
of the ellipse for center, strike an arc cutting the major 
axis produced in the points F and F f . Now take 



-vi 



b 2 



and draw a line DD f (Fig. 15) whose 




Fig. 15. 



equation is x = -. This line will cut the ellipse, because 

a , 
< a. 

e 

Following the analogy of the ellipse, we proceed to 
find the path of a point whose distance from the fixed 
point F is to its distance from the fixed line DD r in 



ANALYTIC GEOMETRY. 129 

a constant ratio, — this ratio e being here defined as 

Va 2 + b 2 

, and hence greater than unity. 

a 

Let P be the moving point whose coordinates are 

x = PN, y = PL. PN cuts DD f at (7, and DD f cuts 

the #-axis at P. 

^ = e; that is, PF=ePC=e(x-^ 
PC \ e 

Also, (PFy = (Pi) 2 + (PL) 2 

Equating the two expressions for (PP) 2 , 

e 2 ix j = y 2 + (x — ae) 2 . 

Expanding and reducing, 

&(# _ 1) _ y 2 = a 2Q e 2 _ ly 9 

a 2 a 2 (e 2 - 1) 
that is, — -|- = 1. 

118. The curve whose equation we have now found 
is the hyperbola. Although closely related to the 
ellipse, it differs from that curve in various important 
respects : 

1. a and b being the same in magnitude and position 
for the two curves, no portion of the hyperbola lies 
within the area occupied by the ellipse ; for as soon as 
x < a, y is imaginary. 



130 CALCULUS. 

2. The hyperbola has two parts or branches sym- 
metrically placed with respect to the axial line in which 
BB' lies ; for if we assign values — x\ — x n , — x ,n , etc., 
to #, we obtain the same values for y that are obtained 
when + x' \ + x n ', + x'", etc., are the values assigned. 

3. Values indefinitely large may be assigned to x 
without making y imaginary. The curve, therefore, 
extends to infinity. 

119. The equation to the hyperbola may be written 

b 



y =± a 



Vx 2 — a 2 ; 



or, expanding (x 2 — a 2 ) 1 by the binomial theorem, 

bf a 2 a* \ , . 

The equations to the two diagonals (produced) of the 
rectangle (Fig. 15) are seen to be 

y = ±-x. (5) 

ct 

Comparing equations (a) and (6), we observe that 
any ordinate of the hyperbola is less than the corre- 
sponding ordinate of the lines ; but we also notice that 
as x becomes larger and larger, the ordinates, according 
to equations (a) and (J), approach equality. 

Whenever such a relation exists between a line and a 
curve, — the distance between them becoming indefi- 
nitely small as the points describing them recede to 
infinity, — the straight line is called an asymptote. 

The hyperbola — — &— = 1 has therefore the asymp- 
7 ct o 

totes y = ± - x. 



ANALYTIC GEOMETRY. 131 

120. The line AA! (Fig. 15), 2 a in length, is called 
the transverse axis of the hyperbola ; and BB f , 2 b in 
length, is the conjugate axis. 

If we consider the equation 

b 2 a 2 

we find it situated with respect to the y-axis just as 
the first hyperbola is with respect to the #-axis. This 

curve, 2- 5=1, is known as the conjugate hyperbola 

b 2 a 2 

in distinction from the primary or transverse hyperbola. 
Following the method of Art. 119, we find that the 

lines y — ±- x are asymptotes of the conjugate 
hyperbola also. 

121. The points A and A! where the transverse axis 
meets the curve are its vertices. Similarly, B and B r 
are the vertices of the secondary or conjugate 
hyperbola. 

The transverse and conjugate axes, the asymptotes, 
and the directrix are all essential lines of the hyperbola, 
and sustain a fixed geometric relation to it like a rigid 
framework, so that if the position of the hyperbola is 
changed with respect to the x- and ^-axes, these lines 
go with it. 

122. If b = a, the rectangle (Fig. 15) becomes a 
square ; the asymptotes become y = ± x, the two lines 
now crossing each other at right angles (Art. 104) ; 
the equation to the hyperbola itself becomes x 2 — y 2 = a 2 , 
and is known as the equilateral hyperbola. It is evi- 



132 CALCULUS. 

dently the hyperbola that would appear in plane M, 
Art. 109, in connection with the circle x 2 + y 2 = a 2 and 
the circumscribed square. 

123. If we rotate the axes in the negative direction 
through the angle — 45°, we shall have these axes coin- 
ciding with the asymptotes of the hyperbola x 2 — y 2 = a 2 . 

To do this we use the formulas 

x = x cos a — y sin tf, 
y = x sin a -f y cos a, (Art. 101) 

which become, for a = — 45°, 

x = x %V2 + y 1V2 = |V2(> + y), 

y = -xW2 + yW2 = W 7 2(y -. x). 

Substituting these values for x and y in the equation 
x 2 — y 2 = a 2 , we have 

that is, i ( .r + y) 2 — \ (y — x) 2 — a 2 , 

which becomes, after reduction, 

a 2 

This is the equation to an equilateral hyperbola 
referred to its asymptotes. 

The isotherm pv — c (Art. 36) is an important illus- 
tration. (See Maxwell's Theory of Heat, Chap. VI.) 

124. Following the method of Art. Ill, and using 
Fig. 15, it is found that 

(FP) 2 = e 2 z' 2 - 2 aex' + a 2 . 



ANALYTIC GEOMETRY. 133 

Taking FP positive, and noticing that ex' > a, we 
have 

FP = ex f - a. 

Similarly, FP = ex 1 + a ; 

therefore FP -FP = 2a, 

and the hyperbola may be defined as the path of a point 
moving so that the difference of its distances from two 
fixed points is a constant. 

125. The polar equation to the hyperbola may be 
readily obtained from Fig. 15. 

Let FP = r, FP = r ! , and the angle LFP = 6. We 
have also r f — r = 2 a and F f F= 2 ae. 

Then r 12 = r 2 + 4 ar + 4 a 2 , 

and since r f is one side of the triangle F'PF, 

r !2 = r 2 + 4 ^2 _ 4 ^ r CQS ( 180 o _ ffy^ 

Equating these two values of r /2 , 

r 2 +■ 4 ar + 4 a 2 = r 2 + 4 a 2 6 2 + 4 a#r cos # ; 

a(e 2 -l) 

hence r = —^ 4- 

1 — e cos # 

This equation may be described as the right-hand 
focal polar equation to the hyperbola. 

When 0=0, r = — a(l + e), and the feather-end of 
the arrow (Art. 45) gives the vertex of the left-hand 
branch of the curve. As the radius vector continues 
to revolve in the positive direction, we continue to get 
negative values of r and points on the left-hand branch 



134 



CALCULUS. 



until 6 = cos l - ; r is then infinite, and the radius 
e j 

vector is parallel to the asymptote y=-x, because 

-il * -\b 

cos i - = tan L - . 

6 a 1 1 

From = cos *- to 6 = 360° — cos -1 -, r is positive, 

and the right-hand branch is being traced. Finally, 
when 6 changes from 300° — cos -1 - to 360°, the re- 
maining part of the left-hand branch is traced. 

126. The parabola. It remains to inquire what kind 
of a curve we have when a point moves so that its dis- 
tance from a fixed point is to its distance from a fixed 
line in the constant ratio unity. 



D 

N 


Y 


c p^- 


B 







X 


i 
D 




\F J 
F] 


r 

[G. 16. 



Let DD\ Fig. 16, be the fixed line, and F the fixed 
point. Let 2p be the length of FB, the distance from 
F to DD f . Take a line through F perpendicular to 



ANALYTIC GEOMETRY. 13 



oo 



DD f for the x-axis, and a line parallel to DD\ bisecting 
FB, for the ^-axis. Let P be the moving point with 
the coordinates OL and PL. 



Then, 


FP = PN=x+p; 


also, 


(FPy = (PL) 2 + (PX) 2 




= y 2 + (x — p) 2 . 


Hence, 


(x + ff = y 2 + (x — p) 2 ; 


that is, 


y 2 = 4 px. 



This curve is called the parabola. 

Since it is the path of a point whose distance from 
the fixed point is to its distance from the fixed line in 
the ratio unity, it must be regarded as the transition 
curve between the ellipse and the hyperbola. Its 
eccentricity 0, being the ratio of the two distances, is 
of course unity. 

127. Considering the equation y 2 = 4px, we see that 
the parabola has a line of symmetry which has been 
used as the a>axis ; for if any value be assigned to #, y 
has two values numerically equal and with opposite 
signs ; so that if the area above the #-axis were folded 
over, the part of the curve in the upper area would 
exactly fit the part in the lower. This line of sym- 
metry is called the axis of the curve, and the point 
where it meets the curve is the vertex. The point F is 
the focus, and the double ordinate through the focus 
is the latus rectum, as in the case of the, ellipse. 

When x—p, y = 2p; therefore the semi-latus 
rectum is twice the distance of the focus from the 
vertex. 



136 CALCULUS. 

128. If p is positive in the equation y 2 = \px, posi- 
tive values may be assigned to x without making y 
imaginary. Hence, the parabola like the hyperbola 
extends to infinity. It differs from the hyperbola, 
however, in this important respect : it has only one 
real branch, because negative values of x make y 
imaginary. 

129. If p is negative in the equation y 2 = \px, we 
have the same law as before, governing the motion of 
the point P ; but the path is now wholly on the nega- 
tive side of the ^/-axis, for only negative values can 
now be assigned to x. 

Similarly, x 2 = 2 py is a parabola above the z-axis, 
with the ?/-axis for its line of symmetry, if p is positive ; 
while x 2 = 2py is the same curve below the #-axis if p 
is negative. 

130. The polar equation to the parabola, the focus 
being pole, is obtained from Fig. 16. FP is r, and the 
angle XFP is 6. Then 

FL = r cos #, 

but r = PJV=FL + 2p; 

therefore, r = r cos 9 + 2 p ; 

that is, r = 2— 



cos 6 

131. The ellipse, parabola, and hyperbola are known 
as conic sections ; for it can be shown that if a cone of 
revolution is cut by a plane in any manner whatever, 
the cross-section is one of these three curves. See 



ANALYTIC GEOMETRY. 137 

Puckle's Conic Sections, Arts. 323-325, together with 
Chap. VIII of that treatise. 

132. A straight line becomes a tangent to a curve at 
any point (x', y'} if (1) it passes through that point, 
and (2) if it has the same slope as the curve at that 
point. We have already seen (Art. 38) that if 

y=zf(£) is the equation to a curve, -^ gives its slope 

dx 

or gradient at each point. It has also been shown 
(Art. 107) that 

y - y ! = ^ O - *0 

is the equation to a straight line passing through the 
point (z f , y'*) with the slope -^. It follows that if 
(x } ', y) is a point P ! on the curve y =f(x) and if -r~ 
is specialized for that point, becoming — ^ or -JL-, 

y - y' = ^7 O - *') 

is the equation to the tangent at P r . 

For example, let us find the equation to the tangent 
at the upper extremity of the latus rectum of the par- 

(a 1J 2 77 

abola y 2 = 4 px. Differentiating, we have -f- = ■— *-. 

ax y 

The coordinates of the upper extremity of the latus 

rectum are #, 2 p. Specializing — ^ for this point. 

dx 

df] = %£ = i . 



138 CALCULUS. 

and the general equation gives 

y - 2p = x - p; 

that is, y = x + p, 

which is the equation to the tangent in question. We 
notice that this particular tangent makes an angle of 
45° with the axis of the curve, which is here the #-axis, 
and cuts the axis produced where the directrix DD f 
cuts it. (See Fig. 10.) 

Example. Find the general equation to the tangent 

to the ellipse ^-+'-^ = 1. Differentiating and solving 

for -^, we have 

(XX 7 70 

ay _ _ b l x 
dx a 2 y 

and the general equation to the tangent becomes, for 
the ellipse, 72 / 

y - y =-- 2 -( x -v)- 

a A y' 

For instance, the coordinates of the upper extremity of 

ty 

the left-hand latus rectum are — ae, bVl — e 2 ; ~ir 

therefore becomes 

b 2, ae 



■ ; that is, e ; 



a 2 bVl- e* 
and the tangent at the point named is 



y — b Vl — e 2 = e(x -f ae). 

It will be noticed that this particular tangent makes 
with the major axis an angle whose tangent is the 
eccentricity. 



ANALYTIC GEOMETRY. 139 

The line perpendicular to the tangent at the point 
of tangency is called the normal. Its equation is 
evidently 

y - y = -j-j( x - x ^ 



133. The general equation to the tangent to the 
parabola y 2 = \px, in terms of the coordinates of the 
point of tangency, is seen to be 

y - y = -6- O - * ) ; 

that is, yy 1 = 2px + y' 2 — 2px ! ; 

or, since y f2 = i px f , (x\ y !> ) being on the parabola, 

W 1 = 2jp(a? + aO, (1) 

and y== ^(aj + ^). (2) 

Writing the equation to the line passing through the 
focus and P(x ] ', y), we have, after reducing, 

If -A is the point where the tangent cuts the axis pro- 
duced, FPA is an isosceles triangle. To prove this, 

2 p 
let tan a= — 4-, the coefficient of # in (2). Then 

y 

tan 2 a = ~^ = 4j?y ' = 4 ^ = _J^_ . 
1 4p 2 y f2 — 4: p 2 ipx'—ip 2 x f — p 

1 ~y¥ 



140 CALCULUS. 

But this is the coefficient of x in (3). 
Hence, angle PAF = angle FPA. 

It immediately follows that if a line Pi is drawn on 
the concave side of the parabola, parallel to its axis, FP 
and PL make equal angles with the tangent.* 

Advantage is taken of this property of the parabola 
in the construction of reflectors. Since the angle of 
reflection of a ray of heat or light equals the angle of 
incidence, if a light be placed at the focus of a para- 
bolic reflector, the light is reflected in a system of 
(approximately) parallel rays. Illumination of a rail- 
road track for a long distance in front of the locomotive 
is secured by means of such a reflector. Conversely, 
if rays of heat or light, parallel to the axis of a para- 
bolic reflector, fall upon its concave surface, they will 
converge at the focus. 

134. It is required to find the locus (path) of the 
middle point of any ellipse-chord moving parallel to 
itself. 

Let C(x n , y rf ) and G\x\ y f ) be the points where 

9 9 

the chord meets the ellipse — + ~ = 1, and let M(x, y) 
bisect the chord CC f . 

* " Hertz, in the first of his celebrated experiments on the propa- 
gation of electric rays, made use of this property of parabolic surfaces. 
He employed large reflectors of sheet zinc bent into the form of para- 
bolic cylinders, in whose focal line the transmitter and the receiver of 
the electric waves were placed. The electric rays passed from the 
transmitter to the first parabolic reflector, were there reflected so as to 
become parallel, and were then reflected from the second reflector to 
the receiver placed at its focus." — Young and Linebarger's Calculus. 



ANALYTIC GEOMETRY. Ill 

Then x = J(V + x fr ), and y = ^(y f + y ,f }; 
that is, x' = 2 x — z rf , and y f = 2y — y lf . 

Since (V, y') and (V', 2/") are each on the ellipse, 

^ + ^ = !> (1) 



£ T + 'V= L ( 2 ) 



Substituting in (1) the values just noticed for x r 
and y f , we have 

(2x-x")* C^y-y^y _ 
a? + b* 

4x 2 -±xx" x" 2 4y 2 -4yy n y n * H 

or I -\ — \- - = 1 

a 2 + a? ^ b* + b* 



Introducing relation (2), 



x(x - x") y(y - y") _ 
a 2 + 62 - u ' 



and therefore ^ „ 



6 2 # 



a; — x n a 2 y 



Now let y — y n = m(x — x ff ) 

be the equation to the chord OC f ; then 



- — Ki = m, 



and this is true, of course, when (#, y), the point tracing 
the line (7(7', is restricted to the point M. 



142 CALCULUS. 



Equating the two values of l ^- rr , 

x — x n 

b 2 x 

a 2 y 

b 2 

that is, y = — - x. 

arm 



The path of the middle point of any chord of slope 

m, moving parallel to itself across the ellipse — + ^- = 1 

a 2 b 2 

is therefore a straight line passing through the center 
of the ellipse. 

If m = 0, the equation becomes x = 0, the equation 
to the minor axis ; and if m = oc, y = 0, the equation to 
the major axis. 

Writing — b 2 for ?> 2 , we have 

b 2 

the corresponding equation in relation to the hyperbola. 

135. By means of the result in the preceding article, 
we are now able to find the center and construct the 
axes of any ellipse. 

Draw any two par?dlel chords and bisect them. The 
chord passing through the points of bisection must 
pass through the center of ellipse ; and the point of 
bisection of this third chord is the center. With the 
center now found and a radius of any convenient length, 
strike a circle cutting the ellipse. Draw one chord 
common to both ellipse and circle, and finally draw 
an ellipse-chord perpendicular to the preceding chord 



ANALYTIC GEOMETRY. 143 

at its middle point. It will be the major (or minor) 
axis of the ellipse. 

Having given the hyperbola with its accompanying 
conjugate hyperbola, the construction of the axes is 
the same as for the ellipse. 

136. The tangent-equation, Art. 132, involves the 
coordinates of the point of tangency. It is desirable 
to obtain a form in which these coordinates do not 
appear. What conditions must be imposed on the line 
y = mx + n so that it shall keep the slope m and yet 
be a tangent to the ellipse —, + — = !? 

Eliminating y between the equations 

y = mx + n, (1) 



x 2 y 2 



a* + P = 1 ' < 2 > 



the resulting equation, 



x 2 {mx + w) 2 _ i 
a 2 b 2 



has for its roots the abscissas of the points of intersec- 
tion of (1) and (2). These roots are 



mn / m 2 n 2 n 2 — b 2 



b 2 b* b 2 



1 m 2 I / 1 m 2 \ 2 1 m 2 

a? ¥ \\a 2 ¥J ~a 2 T 2 

Thus far the straight line is merely a secant (real 

or imaginary) of the ellipse. If it is to become a 



144 CALCULUS. 

tangent, the two points of its intersection with the 
ellipse must be indefinitely near to each other; that 
is, the two abscissas must be equal. Hence, the radi- 
cal, which now makes them unequal, must vanish, and 
we have 

m 2 n 2 n 2 — b 2 



f \ m* Y 1 m 2 

Co 2 W ~a? ¥ 



From this equation of condition we obtain 



n = ± Va 2 m 2 + i 2 , 

which is therefore the relation which must hold between 
n and a, m and b in order that (1) shall be tangent to 
(2), and we have 

y = mx ± ^fa 2 m 2 + b 2 . (3) 

The double sign in (3) plainly means two tangents 
parallel to each other, one cutting the ^/-axis at the 
distance VoW + P above the origin, and the other at 
the same distance below it. 

The corresponding equation for the tangent to the 
hyperbola may be obtained at once by writing — b 2 for 
b 2 in (3), and we have 



y = mx ± Va 2 m 2 — b 2 . (4) 

If b = a, so that the ellipse becomes a circle, (3) 
becomes 

y = mx ± aVm 2 + 1. (5) 

Similarly, if the hyperbola is equilateral, (4) becomes 



y = mx ± a^Jm 2 — 1. (6) 



ANALYTIC GEOMETRY . 145 

Exercises. 
137. 1. Construct the ellipse 

£ + £ = L 

4 9 

What is its eccentricity ? How must the formula for e be 
written in this case ? 

2. Find the points of intersection of the ellipse and 
hyperbola whose equations are 

aj" + 2^ = l, 3x 2 -6f = l, 

and show that at each of these points the tangent to the 
ellipse is the normal to the hyperbola. (Puckle's Conic 

Sections.) 

3. Find the equations to the asymptotes of the hyperbola 
3 x 2 — 6 y 2 = 1. 

4. Find the distance between the right-hand focus of 
x 2 + 2 y 2 = 1 and the right-hand focus of 3xr — 6y 2 = l. 

5. Write the equation to a circle which shall have its 
center coincident with the focus of the parabola y 2 = Ap>x, 
and shall be tangent to the parabola. 

6. Find an expression for the perpendicular distance 

9 9 

XT 1/~ 

from the right-hand focus of the ellipse —^ 4-^=1 to 
the tangent y = mx + Va 2 m 2 + b 2 . 

7. Find an expression for the perpendicular distance 
from the focus of the parabola y 2 = £px to any normal. 

8. Show that the line y = mx + n becomes a tangent to 
the parabola y 2 = 4^px if n = — 

9. Show that the path of the middle point of any parab- 
ola-chord moving parallel to itself is a line parallel to the 
axis of the parabola. 



146 CALCULUS. 

10. Given a parabola, find its axis and focus. 

11. The locus of the foot of the perpendicular from the 
center of the equilateral hyperbola x 2 — y 2 = a 2 is the lem- 
niscate (x 2 + y 2 ) 2 = a 2 (as 2 — y 2 ). 

Use y = mx + a Vm 2 — 1. 

The line perpendicular to it passing through the center 
is y = x. It is required to find the path of the inter- 
section of these two lines as m passes through all values. 
Eliminate m. 

12. Show that the locus of the foot of the perpendicular 
dropped from the focus of the parabola on its tangent is 
the tangent at the vertex. 

13. If a source of light or heat is placed in one focus of 
an ellipse, the rays will be reflected so as to meet in the 
other focus. 

14. A. planet at P is moving in the direction PQ. Its 
distance PS from the sun at S (one focus of its elliptic 
orbit) is J its major axis. Construct the orbit. 

15. Given one focus and any point P and the length of 
the major axis of an ellipse; show that the eccentricity 
depends on the direction of the tangent at P. Construct 
the major and minor axes of ellipses corresponding to 
various tangents through P. 

16. The tangent at any point of a hyperbola is produced 
to meet the asymptotes ; show that the triangle cut off is of 
constant area 

17. Find the equation to the path of the center of a circle 
which is tangent to two given circles. 



ANALYTIC GEOMETRY. 



147 



138. Just as a point in a plane may be determined by 
referring it to two lines at right angles to each other, 
so a point in space may be determined by referring it 
to three planes, each plane intersecting the other two 
at right angles. The point common to the three planes 



z 


• 


M 


/ 











7/^ 


P 


X 








2 


f 



Fig. 17. 

is called the origin, and the lines of intersection of the 
planes are known as the axes of x, y, and z. The posi- 
tive directions of the axes are usually taken to be repre- 
sented in the figure by OX, OY, and 0Z\ the negative 
directions are then OX', OY', 0Z r . 

If P (Fig. 17) is any point in space, then PL, PM, 
PN, its perpendicular distances from the planes YOZ, 
ZOX, and XOY respectively are the coordinates x, y, 
and z respectively. 

139. The three planes evidently divide the space 
around the origin into eight equal triedral angles. If 
a point is in the upper front right-hand angle, its 



148 CALCULUS. 

coordinates are all positive, because each one is meas- 
ured parallel to its own axis and in the positive direc- 
tion. Again, if a point is in the upper front left-hand 
angle, the y and z coordinates are positive, but the 
x coordinate is negative because measured in the nega- 
tive direction parallel to OX' . In like manner we are 
able to state the character of each coordinate for points 
situated in each one of the other six angles. 

140. OP, the distance of P from the origin, is the 
diagonal of the rectangular parallelopiped, three of 
whose edges are PZ, PM, PN. Therefore, 

(OPf = (Pi) 2 + (PJ/) 2 + (PX) 2 
- a? + } f + z i. 

141. Suppose we have any two points P ! (x', y' , z') 
and P" (x" , y" , z"). Let planes be passed through P' 
and P" parallel to the three planes of reference. 
There is thus formed a rectangular parallelopiped 
whose diagonal is the line P f P", and three of whose 
edges are x" — x\ y"—y f , z" — z' . Therefore, 

(jp>p"y = o" - x r ) 2 + o" - y'y + (z" - z'y. 

This formula for the distance between two points in 
space should be compared with the formula in Art. 107. 

142. We have seen (Art. 39) that the general equa- 
tion of the first degree in two variables represents a 
straight line. It may now be asked, What is repre- 
sented by 

Ax + By + Cz + D = 0, (1) 

the general equation of the first degree in three 
variables ? 



AXALYTIC GEOMETRY. 149 

1. It represents a surface and not a solid. For let 
(a, 6) be a point in the plane YOX, and suppose that a 
straight line be drawn through this point parallel to 
the z-axis to meet the locus of equation (1), whatever 
kind of locus it may be. We now have 

-Aa-Bh-D 

z = a ; 

therefore the straight line meets the locus in one 

definite point at the distance — — from the 

plane YOX, and consequently the locus cannot be 
made up of layers either adjacent to one another or 
occurring at intervals. 

2. The surface is a plane. For suppose that a 
point moving in the surface be so restricted that it 
must remain at a constant distance from the plane 
YOX ; that is, let z have a constant value, say c. 
Equation (1) is now reduced to Ax + By + Cc + D = 0. 
Therefore the point moving in the surface and at a 
constant distance from the plane YOX is moving in a 
straight line. In other words, any section of the sur- 
face made by a plane parallel to the plane YOX is a 
straight line. Hence, if Ax + By 4- Oz + D = is not 
a plane surface, it must be a wavy surface, something 
like a corrugated tin roof with the corrugation lines 
parallel to the plane YOX. But repeating the argu- 
ment, making y a constant, we find that all sections 
made by planes parallel to the plane ZOX are straight 
lines. The surface in question must therefore be a 
plane. 

In case the constant term D is zero, the coordinates 



150 CALCULUS. 

of the origin (0, 0, 0) satisfy the equation, and the 
plane passes through the origin. 

Thus the equation, Cz -C'i/ + C"x = (p. 103), 
represents a plane passing through the origin ; and 
any moving point whose coordinates satisfy this equa- 
tion at each instant, must be moving in the plane and 
hence in a plane curve. 

Arts. 138-1-12 have been introduced for the sake of Arts. 86-88. 



dx dx doc 



CHAPTER V. 

FORMULAS. 

li- ( ( ( ^dx + ^dx) = u + v. 
J \dx doc J 

2. J^uv = v^+u dv . a ,. M*—*» + "— *»)=*"' 

doc doc doc i J \ doc doc J 

3. jL u n = nu n - 1 ^ 3,. (u tn —dx= u ™ +1 . 
dx dx J dx m + 1 

4. -4^- sin m = cos w — • 4,. I cos ^ — dx = sin if. 
dx dx J dx 

5. — 'cos w = - sin «« — • 5,. \ sin w — dx - - cos u . 
dx dx J dx 

6. -^- tan ^ = sec 2 u ~- 6,. f sec 2 if ~ *» = tan u . 
dx dx J dx 

7 . — cot u = - cosec 2 m — • 7, . f cosec 2 u — dac=-cotu. 
dx dx J dx 

8. -!^-secw=tanwsecw— • 8,. {t2LMi&eeu^dx=secu. 
dx dx J dx 



9. -7- cosec u = — cot u cosec if — ^- 
ax dx 



, d?f 



9, . 1 cot u cosec tf ^^ dx = — cosec ti. 

dtf du 

d . ! dx _, ^ f dx , . 1 

10. — -Sin-l^rr: — -. 10^ J — - dX = Sin" 1 if. 

ax Vl _ ^2 vi _ ^ 

dtf dw 

11. -^-cos 1 ^ — — Hi. f -^=dx = cos" 1 w. 

dx Vi_^2 J Vf^ 2 

151 



152 CALCULUS. 

du du 

12. -=— tan -1 u - — • 12 L . J - — L — c dx = tan -1 u. 

dx 1 + w a J \ + w 2 

<7?^ dw 

d d ^y* i d nr 

13. — -cot _1 t^=-- 3« 13x. J — - — — ^ da? = cot 1 if. 

da? 1 + u 2 ^ 1 + if 2 

d?f du 

ia d 1 c&e -. , f das , t 

14 - -j— sec _1 i^ = — , 14 x . J doc = sec * n. 



d 1 f/.r 

15. cosec~ * m = 

rca? Mvie- — 1 



d?£ 

15 



J , ' 1 dx = cosec -1 u. 



d ^a? _ ^ac 



16. ^e x = e x . 16 

da? 



17. ^-a r = a x log e «• 17,. f <*> x log" e ada? = a x . 
dx J 

18. -*Le u = e u ^* 18,. fe*^da; = e M . 

da? da? J da? 

19 . <L a « = „«. ^ ioge a . 19l . f «« f« rfx = ^_. 

dx dx J dx log e a 

20. -£- l0ge 05 = X 20:. f — = loge X. 

dx x J x 

du du 

oi d , dx oi f da? ! 

21. ——\0geU = • ^ 1 1" ) —-dx = \0geU. 

dx u J u 

du du 

22. JLvi = -^. 22 1 .J_^^ = VS. 



FORMULAS. 153 

23 . f Vtf^tf dx = ~ sin" 1 - + - Vr?^?. 
J 2 a 2 

24. fVx 2 ±cfdx = \ [xVx 2 ± a? ± a 2 log (x + V# 2 ± a 2 )]. 

25. f __^_ = log (aj + Va; 2 ± a 2 ). 
*^ V# 2 ± a 2 

26. f X dx = + Va 2 4- a 2 . 

J Va 2 ± a 2 

27. f ^ ^ = -^V?3g + ^L sin- 1 ^. 
J Va 2 - z 2 2 ^2 a 

28. f ^ dx = ^Vx T Td 2 T ^ log (a? + Va^dT^ 2 ). 
•^ V# 2 ± a 2 ^ ^ 

29. fxWa^ti 2 dx = ^(2x 2 - a 2 ) V^^ + - sin" 1 -• 
— 7= = - lo g 

#r-\/r/ 2 -4- o^ 2 C6 

. f- 



| — — = ± log g 

«Va 2 ± # 2 a a + Va 2 ± a; 2 

cfa _ Va 2 ± a? 2 



32, ) > Va V*W - Va2 -* 2 -sm-^. 
J x 2 x a 

/X X i ' 

— dx = a vers -1 V2 ax — x 2 . 
■\/9, r/cr. — cr? a 



34. 



/ 



V2 ax — x 2 

dx V2 ax — x 2 



xa/2 ax — x 2 ax 

V 2 ax — x 2 dx = — vers -1 - -\ V 2 ax — x 2 ' 

2 a 2 

36. I — — dec = a vers -1 - + V2 ax — x 2 . 

J x a 



154 CALCULUS. 

37 . I tan x dx = — log cos x. 38. j cot x dx = log sin x. 

39. f-^ = log tan-- 40. f-^-=logtanf- + 

J sin. i- 2 J cos a; \2 4 

41. I - = — cot x. 42. I — — = tan x. 

J suras J cos L # 

C ■ 7 i o sin 2 a? 

43. I sin a; cos a? ao? = — J cos 2 a? = — - — 

44. I — = log tan x. 

J sin x cos x 

45. I x sin .r cfo = sin x — x cos x. 

46. I ar sin x dx = — as 2 cos a; + 2 x sin a; + 2 cos x. 

47 . J sin 2 x dx = — \ sin 2 a: + \ x. 

48. J cos 2 x dx = \ sin 2x + % x. 

49 . I log x dx = a; log a; — a:. 

50. I sin -1 x dx = x sin -1 x + Vl — a?. 

5 1 . f tan" 1 a; c?aj = a; tan" 1 x — \ log (1 + x 2 ) . 
. J sec -1 x dx = x sec -1 a? — log (x + Va? + 1) . 

. i fa>6 , f f = -| = tan-^J^tan| 

J Ja+6cosx Va 2 -6 2 V Xa + & 2 



52 



For other integrals see Peirce's Short Table of Integrals. 



FORMULAS. 155 

54. f(z + x)=f{z)+f(z)x+f^ + £^^+.... 

55. f(x) =/(Q) +/' (O) X + £^j^- + / "^°) * + .... 

56. (a + rt)" = a" + ma-'x + !il!izll^ 

m (m — 1) (m — 2) a" 1 " 3 a^ 

57. e ^l + , + | + | + .... 

58. a- = l + g ; log e a + x2(log ' a)2 + g:3(1 ^ a)3 + -.. 

[2 [3 

/y»2 /ytO /y»4 

59. log(l + a; ) = x-| + |-|+.... 

60. sm^ = ^-, — h, ■••- 61 • cos# = l — ■ — K ■•• 

[3 16 |2 li 



62. sin ( - + « ] = cos a. 63. cos [ - + a ) = — sin a. 

64. tan (- + «]=— cot a. 65. cot (5 + a] = — tan a. 

66. sin (— a) == — sin a ; cos (— a) = cos a. 

67. tan (— a) = — tan cc ; cot (— «) = — cot a. 

• o o -i ™ *. sin a sin a 

68. sm- « + cos 2 a = 1. 69. tan a = 



cos a Vl - sin 2 a 

70. sin (a + /?) = sin a cos /2 + cos a sin /3. 

71. cos (a + /?) = cos a cos /J — sin a sin /?. 

72. sin (a — /?) = sin a cos /? — cos « sin /?. 

73. cos (a — j8) = cos a cos /? + sin a sin (3. 

74. sin 2 a = 2 sin a cos a. 



156 CALCULUS. 

75. cos 2« = cos 2 a — sin 2 a = 2 cos 2 a — 1 = 1 — 2 sin 2 a. 

„„ , / , m tan ft + tan /? 

76. tan (a + /?) = — • 

1 — tan « tan /? 

„ . / n \ tan « — tan B 

77. tan (ft — S) = ^-« 

1 + tan ft tan /3 

78. tan2«= 2tan " ■ 79. cot 2 a = COt2 <* ~ 1 



1 — tan- ft 2 cot < 



ft /l — COS ft - ft /l 

80. Sm =yj- 81. COS 2 = > /- 



™ , ft /l — COS ft on , ft /1.+ COS ft 

82. tan-=\- 83. cot-=\/— ?- 

2 \ 1 + cos « 2 \ 1 — cos ft 

84. sin « + sin /3 = 2 sin i (ft -f- /5) cos i (ft — /}). 

85. sin « — sin f3 = 2 cos -J (ft + /?) sin J (ft — /3). 

86. cos « + cos /} = 2 cos £ (ft + /J) cos | (ft — /J). 

87. cos ft — cos (3 = — 2 sin J (ft + /J) sin \ (a — /?). 



88. log afr = log a + log 6. 89. log - — log a — log b. 

b 

1 i 

90. log a n = n log a. 91. loga n = -loga. 

n 

92. If asc 2 + bx + c = 0, 

& F& 2 c & 1 m — r 

2 a \4 a 2 a 2a 2a 

93. [* = l-2?3'-4 — n. 94. log 1=0. 

95. Iog0=— oo. 96. log a a=l. 

97. e = 2.7182818284 .... 98. log 10 e = 0.43429448 •• 

99. 7T = 3.14159265-... 100. log 10 7r = 0.49714987 •■ 



FORMULAS. 157 



101. #° = — = 57°.2957795 



102. R" = 180 ' 60 ' 60 " = 206264".8 •• 



103. log R° = 1.75812263 • •-. 

104. log R" = 5.31442513 •••. 

105. 30° 45° 60° 
sin, | iV2 |V3 
cos, iV3 |V2 i 

106. Base of right triangle = /^ cos y ; alt. = h sin y. 

Qi = hypotlienuse ; y = angle at base.) 

107. Area of sector of circle = \ r(rO) = \ r*9. 

108. Area of ellipse =wab. 



INDEX. 



The references are to pages. 



Abscissa, 42. 
Acceleration, 62. 
Amplitude, 67. 
Analytic geometry, 111. 
Anomaly, true, 107. 
Aphelion, 106. 
Apsides, 57. 
Archimedes, 2. 
Areal velocity, 105. 
Areas, 85. 

of surfaces of revolution, 94. 
Asymptote, 130. 
Asymptotes of hyperbola, 130. 
Attraction of homogeneous sphere, 

70. 
Axes of coordinates, 41. 
Axes, change of, 115. 

major and minor, 122. 

transverse and conjugate, 131. 
Axis of parabola, 135. 

of symmetry, 135. 

Binomial theorem, 30. 
Bodies, falling, 67. 
Boyle's law, 2. 

Cartesian coordinates, 41, 111. 
Catenary, equation to, 93. 
Circle, equation to, 46. 



Circular functions, 20. 
Comets, orbits of, 108. 
Concavity of curves, 48. 
Conic sections, 136. 

exercises on, 145. 
Conjugate hyperbola, 131. 
Constant, derivative of, 8. 

of integration, 13. 
Convergence of series, 30. 
Coordinates, 42. 

current, 43. 

polar, 56. 

transformation of, 56, 
115. 



114, 



Definite integrals, 14. 
Derivative, 5. 

of constant, 8. 

of product, 9. 

of quotient, 15. 

of sum, 8. 

of cos x and sin x, 18. 

of x n , 10. 
Derivatives, second and higher, 
24. 

partial, 24. 
Descartes, 111. 
Differential, 5. 

perfect, 98. 

total, 25. 



159 



160 



INDEX. 



Differentiation, 8. 
Directrix of ellipse, 126. 

of hyperbola, 129. 

of parabola, 138. 
Displacement, 67. 
Distance between two points, 119. 

from point to line, 119. 
Double integrals, 97. 

Eccentricity of ellipse, 123. 

of hyperbola, 129. 

of parabola, 135. 
Ellipse, 121. 

area of, 87. 

axes of, 122. 

construction of, 125. 

determination of center and 
axes of, 142. 

directrix of, 126. 

eccentricity of, 123. 

ellipticity of, 123. 

equation to, 122. 

path of middle point of chord 
' of, 140. 

polar equation to, 127. 

foci of, 123. 

vertices of, 122. 
Energy, kinetic, 92. 
Epoch. 67. 
Equation to a curve, 111. 

ax + by + c = 0, 45. 

Ax + By + Cz + D = 0, 148. 
Equations of motion, 78. 
Equilateral hyperbola, 131. 

Falling bodies, 67. 
Foci of ellipse, 123. 

of hyperbola, 128. 
Focus of parabola, 135. 



Formulas, collection of, 155. 
Function, algebraic, 4. 

explicit and implicit, 16. 

periodic, 47. 
Function, transcendental, 4. 

of several variables, 24. 

Gradient, 44. 
Graph, 41. 

Gravitation, law of, 2. 
Gravity, 67. 

Harmonic motion, 67. 

Heat, minimum intensity of, 54. 

Hertz, 140. 

Horizontal range, 79. 

Hyperbola, 127. 

asymptotes of, 130. 

axes of, 131. 

directrix of, 129. 

eccentricity of, 129. 

equation to, 129. 

polar equation to, 133. 

foci of, 128. 
Hyperbola, conjugate, 131. 

equilateral, 131. 

transverse, 131. 

Indeterminate forms, 31-33. 
Indicator diagram, 97. 
Inertia, moment of, 100. 
Inflexion, point of, 49. 
Integrals, double and triple, 97. 

definite and indefinite, 13. 

table of, 151. 
Integration, 12. 

constant of, 13. 

by parts, 37. 
Intercept, 119. 
Isotherm, 132. 



INDEX. 



161 



Kepler's laws, 102. 
Kinetic energy, 92. 
of rotation, 99. 

Latus rectum, 123. 
Law, Kepler's first, 107. 

Kepler's second, 105. 

Kepler's third, 110. 
Lengths of curves, 92. 
Limits of definite integrals, 14. 
Locus, 41. 

Logarithmic differentiation, 21. 
Logarithms, common and Napier- 
ian, 39. 

Maclaurin's theorem, 29. 
Maxima and minima, 48. 

exercises in, 52-55. 
Mean values, 88. 
Moment of inertia, 100. 
Motion, equations of, 78. 

of falling body, 68. 

of rising body, 69. 

in a parabola, 77. 

pendulum, 82. 

rectilinear, 74. 

simple harmonic, 67. 

in a vertical curve, 81. 

Newton, 2. 

Node, longitude of ascending, 116. 

Normal, 139. 

Ogee, 49. 

Operation, indicated, 12. 
Orbits, eccentricity of, 108. 
Ordinate, 42. 
Origin of coordinates, 41. 
change of, 114. 



Parabola, 134. 

axis of, 135. 

directrix of, 138. 

eccentricity of, 135. 

focus of, 135. 

latus rectum of, 135. 

equation to, 135. 

polar equation to, 136. 

vertex of, 135. 
Parabolic reflector, 140. 
Parallelism, condition of, 117. 
Partial differential coefficients, 24. 
Pendulum, 82. 

time of oscillation of, 84. 
Perfect differential, 98. 
Perihelion, 106. 
Period, 67. 
Periodic function, 47. 

time, 110. 
Perpendicularity, condition of, 

117. 
Phase, 67. 

Point of inflexion, 49. 
Polar coordinates, 56. 
Pole, m. 
Projectile, path of, 78. 

Radian, 47, 85. 
Radius vector, 57. 
Range, horizontal, 79. 

on an incline, 80. 
Revolution, areas of surfaces of, 
94. 

volumes of, 94. 

Sinusoid, 47. 
Slope, 44. 
Straight line, 117. 

exercises in, 120. 
Symmetry, line of, 135. 



162 



INDEX. 



Table of formulas, 155. 

of integrals, 151. 
Tangent in terms of slope and 
intercept, 143. 

to curve, 137. 
Taylor's theorem, 28. 
Transformation of coordinates, 

56, 114, 115. 
Transverse hyperbola, 131. 
Triple integrals, 97. 



True anomaly, 107. 

Variable, 3. 
Velocity, areal, 105. 

angular, 63. . 

component, 60. 

linear, 58, 64. 
Vertices of ellipse, 122. 
Volumes of revolution, 94. 

Work, 91. 



SCIENCE. 51 

Physics for University Students. 

By Professor HENRY S. CARHART, University of Michigan. 

Parti. Mechanics, Sound, and Light. With 154 Illustrations. i2mo, 

cloth, 330 pages. Price, $1.50. 

Part II. Heat, Electricity, and Magnetism. With 224 Illustrations. 

i2mo, cloth, 446 pages. Price, $1.50. 

THESE volumes, the outgrowth of long experience in teach- 
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not a cyclopaedia of physics. 

Particular attention has been given to the arrangement of 
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ing the relations of physical quantities. At the same time the 
course in Physics represented by this book is supposed to pre- 
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prepare the student for the study of higher mathematics. 

Professor W. LeConte Stevens, Rensselaer Polytech?tic Institute, Troy, N. Y. : 
After an examination of Carhart's University Physics, I have unhesitat- 
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comes nearer meeting my special needs than any book I have examined, 
being far enough above the High School book to justify its name, and 
yet not so far above it as to be a discouragement to the average student. 



52 SCIENCE. 

Primary Batteries. 

By Professor HENRY S. CARHART, University of Michigan. Sixty- 
seven Illustrations. i2mo, cloth, 202 pages. Price, $1.50. 

THIS is the only book on this subject in English, except a 
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They are free from bias and exhibit some facts not heretofore ac- 
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The battery as a device for the transformation of energy is; 
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late you on producing a work which contains a great deal of information 
which cannot be obtained readily and compactly elsewhere. 



SCIENCE. 53 



Electrical Measurements. 

By Professor Henry S. Carhart and Asst. Professor G. W. PATTER- 
SON, University of Michigan. i2mo, cloth, 344 pages. Price, $2.00. 

IN this book are presented a graded series of experiments for 
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The Electrical Engineer, New York : We can recommend this book very 

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The Electrical Journal, Chicago: This is a very well-arranged text-book 

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Exercises in Physical Measurement 

By Louis W. Austin, Ph.D., and Charles B. Thwing, Ph.D., 
University of Wisconsin. i2mo, cloth, 198 pages. Price, $1.50. 

THIS book puts in compact and convenient form such direc- 
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Introduction with Part II. gives a very clear presentation of the essential 
things in Measurements, and of the treatment of errors. 



58 MATHEMATICS. 



Elements of Algebra. 

By Professor James M. Taylor, Colgate University, Hamilton, N.Y. 
At Press. 

IN this book Professor Taylor aims primarily at simplicity in 
method and statement, and at a natural and logical sequence 
in the series of steps which lead the pupil from his arithmetic 
through his algebra. 

An introductory chapter explains the meaning and object of 
literal notation, and illustrates the use of the equation in solving 
arithmetical problems. This is followed by a drill on particular 
numbers before the pupil is introduced to the use of letters to 
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brought out by induction from particular cases, and proofs are 
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memorize without comprehending them. Nomenclature has been 
looked to carefully. Many of the misleading terms of the older 
text-books have been discarded and others more useful and help- 
ful have been applied. 

The methods of working examples have been chosen for their 
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method of attack are given, but formal rules are stated but rarely. 
Positive and negative numbers are so explained and defined as to 
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the statement of rules. 

The book is particularly adapted to beginners, and is intended 
at the same time to prepare for any college or scientific school, 
as each subject is so treated that the pupil will have nothing to 
unlearn as he advances in mathematics. 



MA THE MA TICS. 59 



An Academic Algebra. 



By Professor J. M. Taylor, Colgate University, Hamilton, N.Y. i6mo, 
cloth, 348 pages. Price, $1.00. 

THIS book is adapted to beginners of any age and covers 
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Factoring is made fundamental in the study and solution of 
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Professor C. H. Judson, Fur?7ian University, Greenville, S.C. : I regard 
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I like the treatment of the theory of limits, and think the student should 
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Logarithmic and Other Mathematical Tables. 

By William J. Hussey, Professor of Astronomy in the Leland Stan- 
ford Junior University, California. 8vo, cloth, 148 pages. Price, $1.00. 

IN compiling this book the needs of computers and of students 
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60 MA THE MA TICS. 



A College Algebra. 



By Professor J. M. TAYLOR, Colgate University, Hamilton, N.Y. 
i6mo, cloth, 326 pages. Price, $1.50. 

A VI GO ROUS and scientific method characterizes this book. 
In it equations and systems of equations are treated as 
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lytical growth. Of course, no book can replace the clear-sighted teacher ; 
for him, however, it is full of suggestion. 



10V 3" 1900 



